Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional assumption $f$ continuous and what ...
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0answers
16 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
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0answers
9 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in ...
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1answer
11 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
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0answers
14 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
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0answers
9 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
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1answer
41 views

homogeneous differential equations $y' = f(y/x)$

There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition). I was unable to prove it, or to find a proof to it. Can any one ...
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1answer
23 views

prove $\sup X \le \sup Y$ if $X$ is a subset of $Y$ both sets are nonempty and $Y$ is bounded above

Working through foundations of mathematical anlysis by johnsonbaugh per suggestion and wondering if the following proof works? (no solutions to book) Problem: Let X and Y be nonempty subsets of real ...
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1answer
12 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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0answers
12 views

Runge's Theorem Application

Below is a question out of Gamelin's Complex Analysis which I cannot quite figure out. Any tips would help appreciated! "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that accumulates ...
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1answer
38 views

About the proof of the four colour theorem

The proof of the classification of finite simple groups is thousands of pages which does not seem to be a human readable proof because its too long (more than 10000 pages) to be read by one person ...
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2answers
50 views

Proof by induction: $(a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+…+nab^{n-1}+b^n$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $a$ and graded for ...
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2answers
40 views

ELI5: Riemann-integrable vs Lebesgue-integrable

I am wondering what the difference between riemann-integrable and lebesgue-integrable means. Does it have anything to do with the absolute value of the integrand, something like ...
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0answers
45 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
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0answers
14 views

Volume Problem in Munkres' analysis on manifolds

I am having trouble with problem (a) of this question. I figured that the volume of $\triangle_1(R)$ is $|(\alpha(a+h, b)-\alpha(a, b))\times (\alpha(a+h, b+k)-\alpha(a, b))|$ but don't know how I ...
3
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1answer
56 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
2
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2answers
81 views

Theorem 2.43 in Baby Rudin: How to understand the proof?

Here's Theorem 2.43 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Here's the ...
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0answers
32 views

Show that $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a complex number)

How do I prove this? Suppose that $a, b$ and $c$ belong to $\mathbb C$ and that $$\lim_{z\to z_0} f(z)=a$$ and $$\lim_{z\to z_0} g(z)=b.$$ a - $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a ...
2
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2answers
36 views

Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
3
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1answer
55 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
0
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1answer
38 views

How to bound this complex number from below?

I am doing an $\epsilon-\delta$ proof ($z \rightarrow i, f(z) \rightarrow \infty$) and currently have the absolute value $$|f(z)|=\left|\frac{z-1}{z^2+1}\right|$$ and I wish to make a statement about ...
2
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5answers
33 views

Prove a function is in Big-Oh and not in Big-Omega

We are told to use the definitions of Big-Oh and Big-Omega to prove that a given function is in $O(f(n))$ or $\Omega(f(n))$. It requires being able to use $c$ and $n_0$. Use the definitions to show ...
3
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2answers
32 views

Composition involving bounded linear operators

I recently come across the following statement mentioned in a proof: Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y ...
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0answers
26 views

Convex function from condition

Can we deduce that $F$ is a convex function (i.e $F''(t)>0, \forall t>0$) from the following conditions: $F(t)=\int_{0}^t f(\xi) d\xi$, $0\leq \theta F(t)< t f(t), \forall t>0$ The ...
2
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3answers
59 views

Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist?

Why is this limit non-existant? $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ I can't seem to find $2$ different paths that would show it is non-existant.
0
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1answer
11 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
2
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1answer
38 views

How would you approach the limit $\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$? [on hold]

How would I solve the limit $\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$ ? Note that $p \in [1,\infty]$.
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3answers
59 views

Is there anything wrong with this definition of discontinuity?

Is there anything wrong with this definition of discontinuity for a function y = f(x)? $\forall \delta>0\, \exists \varepsilon>0$ such that $\vert x-c\vert < \delta$, but $\vert f(x) - ...
0
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1answer
40 views

How to prove $\displaystyle\bigcup^\infty_{k=1}(\bigcap^\infty_{n=1}A_{k,n})\subset\bigcap^\infty_{n=1}(\bigcup^\infty_{k=1}A_{k,n})$

Want to show $$\displaystyle\bigcup^\infty_{k=1}\left(\bigcap^\infty_{n=1}A_{k,n}\right)\subset\bigcap^\infty_{n=1}\left(\bigcup^\infty_{k=1}A_{k,n}\right)$$ Note the bottoms are $k=1,n=1$ and ...
3
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1answer
40 views

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
5
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7answers
141 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
1
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1answer
22 views

intersection of intervals from 1 to infinity

This is a continuation of my previous quesion concerned with finding $ \cup^\infty_{k=1} S_k $ for $ S_k = (1 − 1/k, 2 + 1/k], k \in N $. And the answer intuitively makes sense, but what about for ...
0
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2answers
24 views

proving union of interval equals interval

Consider the interval $ S_k = (1 − 1/k, 2 + 1/k] $ and I want to find $ \cup^\infty_{k=1} S_k $ with proof How do I go about this? What I was thinking... as $ k \rightarrow \infty $ $ 1/k ...
2
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2answers
23 views

Supremum not in interior of set $A$

If $ A \neq \emptyset $ and is bounded above and $ s = \sup A $ then $ s \in \bar A $ and $ s \notin \operatorname{int}A$ I've got the first part, showing that $s$ is a limit point of $A$. Thus, $ s ...
0
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1answer
22 views

Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
0
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1answer
21 views

How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
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3answers
29 views

Why is $\sup_{x∈[0,1]} {|p'(x)|} ≤ A_d\sup_{x∈[0,1]}{|p(x)|}$ for all polynomials $p$ of degree at most $d$?

How can one prove that for any positive integer $d$, there is a constant $A_d < 0$ such that $$ \sup_{x∈[0,1]} {\lvert\, p'(x)\rvert} ≤ A_d\sup_{x∈[0,1]}{\lvert\, p(x)\rvert}, $$ for all ...
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0answers
17 views

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$?

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$, which is finite and of degree at most $d$ ? Continuity of a function ...
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2answers
17 views

Why do the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$ lie in ${R}^{d+1}$'s unit sphere?

Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$. Why would it suffice to suffices to assume that this vector lies in the unit ...
0
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1answer
24 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
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0answers
18 views

Proof of Well-ordering principle [on hold]

In analysis class we are building natural numbers from the real numbers The teacher gave us a Theorem Theorem(Well-ordering principle) Let $G \subseteq \Bbb{N}$ non-empty then $\exists m_0$$\in G: ...
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0answers
25 views

Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded

Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
2
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1answer
16 views

Continuity on the parameters of the intermediate value theorem

Let $X$ be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and $F : X \times [0, 1] \to \mathbb{R}$ a continuous function such that $F(x, 0) ...
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votes
0answers
22 views

Suppose the MRL of a continuous survival time X is given by MRL(x)=x+10. Find the mean of x and both the hazard and survival functions.

For finding the mean of X, I simply plugged 0 into x and got 10 as a result knowing that the mean can be found by mrl(0). I know the basic formulas for hazard and survival functions but I do not ...
2
votes
1answer
54 views

Show $A=\limsup_\limits{n\to\infty}a_n$.

Let $\{a_n\}$ be a sequence of real numbers bounded from above, $A\in \Bbb R$. Given any $\epsilon>0$, a)$\exists n_0 \in \Bbb N$ such that $ a_n<A+\epsilon$ for all $n\ge n_0$. b)$\exists ...
1
vote
1answer
25 views

Constant $a_{k}$ which normalizes integral is bounded

For $x \in [-\pi,\pi]$ define $$f_{k}(x) = a_{k}\cdot \left(\frac{1+\cos(x)}{2}\right)^{k}$$ where $a_{k}$'s are chosen such that $$\frac{1}{2\pi} \cdot \int_{-\pi}^{\pi} f_{k}(x) \ dx =1 $$ Then show ...
0
votes
4answers
45 views

How can we calculate the limit at $0$?

Suppose we have the function $g:\mathbb{R}\rightarrow \mathbb{R}$ and $g(x)=\left\{\begin{matrix} x\cdot (-1)^{[\frac{1}{x}]} & x\neq 0\\ 0 & x=0 \end{matrix}\right.$ I want to show that ...
1
vote
2answers
23 views

How do we show the limit?

We have that $(a_n)$ is a bounded sequence of real numbers that satisfy $2a_{n+1}\leq a_n+a_{n+2}, \forall n\in \mathbb{N}$. I want to show that the sequence $b_n=a_{n+1}-a_n$ converges and that the ...
-1
votes
3answers
38 views

How do you show a set is dense? For example, is the set of all ration numbers $p/q$ with $q ≤ 10$ a dense set?

How do you show a set is dense? For example, is The set of all ration numbers $p/q$ with $q ≤ 10, p \in \Bbb Z, q \in \Bbb N$ a dense set? I know that a set is considered dense in $\Bbb R$ if an ...
4
votes
2answers
72 views

Prove that if $\sum_{n=1}^ \infty na_n$ converges, then $\sum_{n=1}^ \infty a_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty na_n$ converges, then $\displaystyle\sum_{n=1}^ \infty a_n$ converges. No, $a_n$ are not necessarily positive numbers. I've been trying summation by ...