Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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124 views

Establish the convergence or divergence of a sequence [duplicate]

Establish the convergence or divergence of the sequence (y_n), where: y_n := 1/(n+1) + 1/(n+2) + ... = 1/(2n) for n /in N.
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1answer
51 views

Assuming continuity of $f$, how does one prove uniform continuity of $f$, assuming the limit to $+\infty$ and $-\infty$ is $0$?

Assuming continuity of $f$, how does one go about proving uniform continuity of $f$, assuming the limit to $+\infty$ and $-\infty$ is $0$? Note: $f\colon\mathbb R\to\mathbb R$. Looking for hints, ...
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0answers
24 views

Open Sets Proof

Suppose that $A,B ⊆ ℝ$. Prove that if $A$ and $B$ are open, then (i) $A \cup B$ is open, and (ii) $A \cap B$ is open. Workings: Proof: (i) Let $x \in A \cup B$ Must find a neighbourhood $y$ of ...
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2answers
99 views

Prove that the limit of the following function of two variables is zero

I need to prove the following: $$\lim_{(x,y)\to (1 ,2)} \frac{x^2+2xy-6x-2y+5}{\sqrt{(x-1)^2+(y-2)^2}}=0$$ I've tried to solve it by substituting $y=mx$ but I can't get the solution that way. ...
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26 views

find the supremum of a set

Let $x \in \mathbb{R}$ with $0 < x < 1 $. Prove $s_n = \sum_{k=1}^n x^k $is bounded and find $\sup \{ s_n : n \in \mathbb{N}\} $. Atempt We have that $0 < x < 1 \implies x^k < 1 $ ...
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29 views

Lateral derivative related with integrability at a point in a interval.

Let $f:[a,b] \to \mathbb{R} $, continuous from right at $ x_0 \in [a,b) $. Proof that $ F:[a,b] \to \mathbb{R} $, defined as $ F(x) = \int_a^x f(t) dt$, is derivable at right of $x_0$, with $ ...
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1answer
31 views

Supremum of a non-decreasing sequence is the limit of the sequence

Suppose $(s_n)$ is a non-decreasing sequence of real numbers. Prove that $\sup\{ s_1,s_2,....\} = \lim s_n $ Attempt: Suppose $A = \lim s_n$. We show $A = \sup \{ s_n \} $. Let $\epsilon > 0$ be ...
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2answers
36 views

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$ where $|z|<1$ and $k \geq 0$. I know The right hand side: \begin{align*} \frac{1}{(1-z)^{k+1}} & = ...
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3answers
41 views

Find the limit of $x_n = \sin(2\pi (n^3 − n^2 + 1)^{1/3} )$

Show that the sequence $(x_n)_{n\geq 1}$ defined by $x_n = \sin(2\pi (n^3 − n^2 + 1)^{1/3} )$ is convergent and compute its limit. What I tried: Let us write $c_n = n^3 - n^2 + 1$. Then we can see $$ ...
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29 views

Exponential decay function multiplied by a lineair function

I have the following two function, one is exponential and the other linear. When i multiply them shouldn't there be an optimum? (So basically a minimum)
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2answers
45 views

Prove: $a_n \leqslant x$ $\forall_{n \geqslant 1} \Rightarrow lim_{n \to \infty}$ $a_n \leqslant x$

I'm not sure about how to do the proof of this exercise of my math study. Its exercise 5.4.8 of Analysis I by Terence Tao: Let $(a_n) _{n=1}^{\infty}$ be a Cauchy sequence of rationals and $x \in ...
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2answers
41 views

Is every convergent sequence in $L^1$ dominated?

This looks very obvious, but I can't prove it, and google is not helping (I also can't find any explicit mention to this in my textbook). I want to prove that, given a measure space $\left(X,\mathcal ...
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1answer
108 views

Examine whether the identity map is continuous or NOT.

Consider the space $C[0,1]$. Consider the following metrics $$d_1(f,g)=\int_0^1|f(t)-g(t)|\,dt.$$ $$d_2(f,g)=\left(\int_0^1|f(t)-g(t)|^2\,dt\right)^{1/2}.$$ $$d_3(f,g)=\sup_{0\le t\le ...
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1answer
45 views

Finding a fixed point of a function satisfying $|f(x)-f(y)|\le K\cdot |x-y|$.

Let $f:[a,b]\to [a,b]$ fulfill $|f(x)-f(y)|\le K\cdot |x-y|$ $\forall x,y \in [a,b]$, $0<K<1$. Prove $\exists! z\in [a,b]$ such that $f(z)=z$. I am really lost here. I would appreciate your ...
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2answers
30 views

Absolute Value Brackets Proof With Epsilon

Prove that if $|x−y| < \epsilon$ for all $\epsilon > 0$, then $x = y$. Workings: Proof: Suppose for a contradiction that $|x−y| < \epsilon$ for all $\epsilon > 0$ and $x \neq y$. Then ...
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0answers
25 views

Why is the following sequence of equalities true?

I am trying to understand the answer posted in composition of two measurable function. Note that $$(\chi_A \circ \Psi^{-1})^{-1}((.5,1.5)) = (\chi_A \circ \Psi^{-1})^{-1}(\{ 1 \}) = ...
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1answer
95 views

Interchange summation and differentiation for ONB

Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that $$\frac{d}{dx}e_n = \lambda_n e_n.$$ Assume $f \in H^1[0,1],$ so i.e. $||f'||_{L^2} < \infty$ ...
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1answer
26 views

How does the intermediate value property differ from the value theorem

IVP: A function $f$ has the intermediate value property on an interval $[a,b]$ if for all $x < y$ in $[a,b]$ and all $L$ between $f(x)$ and $f(y)$, it is always possible to find a point $c \in ...
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5answers
114 views

Show that $f''(x) = 0$ for some $x$

Let $f$ be a twice differentiable function with the following properties: $f(x) > 0$ for $x \ge 0$. and $f$ is decreasing, and $f'(0) = 0$. Prove that $f''(x) = 0$ for some $x > 0$. The ...
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0answers
28 views

Necessary conditions on $x$ for $e^{ax} \geq b - cx$.

Let $a,c>0$, $b>1$ be constants. I am wondering what kind of necessary conditions we can find in order that $$ e^{ax} \geq b - cx$$ holds. So I would like something like $$ e^{ax} \geq b ...
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1answer
13 views

Proof of a polynomial given parameters

Let $a_1, a_2, ... a_n$ and $b_1, b_2, ... b_n$ be given numbers. If $x_1, x_2, ... x_n$ are distinct numbers, prove that there is a polynomial function $f$ of degree $2n - 1$, such that $f(x_j) = ...
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67 views

Computability, Continuity and Constructivism

Triggered by an IMO extremely interesting question & Mathematics Stack Exchange, asked by Dal: Computability and continuous real functions And a link in one of the comments that could have ...
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1answer
62 views

Bounding $|f'|$ given a bound for $|f|$ and $|f''|$. [duplicate]

I came across this problem on a Berkeley preliminary exam, and have yet to come up with a solution. Suppose that $f$ is a twice-differentiable real-valued function on the real line such that ...
2
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1answer
22 views

Compactness of translation operator in weighted spaces

Let $x,v\in\Bbb R^d$, $t\in \Bbb R$ and $m(x,v)$ be a smooth strictly positive function rapidly decaying on infinity - think $m(x,v) = \exp(-|x|^2-|v|^2)$. Define Banach spaces $X$ and $Y$ by ...
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1answer
23 views

$f$ is a real valued function defined for all real numbers $x$ such that $0\le f(x)\le \dfrac12$ and…

$f$ is a real valued function defined for all real numbers $x$ such that $0\le f(x)\le \dfrac12$ and for some fixed $a>0$, $f(x+a)=\dfrac12-\sqrt{f(x)-(f(x))^2}$ for all $x$. Show that the function ...
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2answers
55 views

Consider $A=\bar B\subset C[0,1]$. Then which are TRUE?

Consider $A=\bar B\subset C[0,1]$ , where, $$B=\left\{f\in C^1[0,1] :|f(x)|\le 1,|f'(x)|\le 1,\forall x\in [0,1]\right\}.$$ Then, $A$ is (a) Closed. (b) Connected (c) Compact (d) Dense. My ...
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4answers
131 views

How does Volume work with integration?

Using a cross section suppose, as described here: Area formula Paul Notes Suppose this is: $y = f(x)$. He says the volume is: $$\int_{a}^{b} A(x) dx$$ But how does area over that interval ...
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3answers
67 views

Finding a sequence of sets whose intersection is a null set

Find a sequence of sets $I_n=\{r:r \in \mathbb{Q}, a_n\le r \le b_n\} $ in $\mathbb{Q}$, where $a_n, b_n \in\mathbb{Q}$ such that $$I_{n+1} \subset I_n\forall n\in\mathbb{N}$$ $\lim_{n \to ...
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1answer
42 views

Stone Weierstrass: Modulus

Given the Consider the modulus: $$M(t):=|t|\quad(-1\leq t\leq1)$$ Introduce the square root: $$R(s):=\sqrt{1-s}\quad(0\leq s\leq1)$$ They are related by: $$M(t)=R(1-t^2)$$ Regard the binomial ...
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2answers
41 views

Show that $B((a,b),r) \subset A\times B$

Question: Let $A$ and $B$ be non-empty subsets of $\mathbb{R}$. Prove that $A \times B$ is an open subset of $\mathbb{R}^2$. NOTE: Fellow M.SE users pointed out that the question lacks ...
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0answers
109 views

Sobolev norm of a convolution

Let $\eta$ be a rapidly decaying function such that it is radial and $(\mathscr{F}\eta)(\xi)=1$ for $\vert\xi\vert\leq 1$. (Here $\mathscr{F}$ is the Fourier transform). Let's put ...
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0answers
19 views

Basic problem on flow

A semiflow $\phi$ on a metric space $(M,d)$ is a continuous map $$ \phi: \mathbb{R}_+ \times M \to M,\\ (t,x) \to \phi(t,x) = \phi_t(x)$$ s.t. $\phi_0$=Identity, $\phi_{t+s} = \phi_t \circ \phi_S$ ...
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3answers
224 views

Prove : $|x - y| \leqslant \epsilon$ $\forall \epsilon > 0$ iff $x = y$

I have to do this exercise for my math study, and I'm having trouble with doing the second part of it. Let $x, y, \epsilon \in \mathbb{R}$ and $\epsilon > 0$. Prove: $|x - y| \leqslant \epsilon$ ...
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1answer
35 views

Interior Point of $ A = \{ \frac{1}{n} : n = 1, 2, 3, … \}$

I have stumbled upon this question in a text book, where one is expected to find the set of interior points of this given set in the real line with the usual metric. I feel like this set doesn't have ...
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1answer
22 views

How to show $\{(x,y)\in \mathbb{R^2}: px+y=1\}$ is unbounded?

I need to show that the set $D=\{(x,y)\in \mathbb{R^2}: px+y=1\}$ is unbounded, where $p>0$ I know this means that I need to show that for all $M>0$, there exits $(x,y)\in D$ such that ...
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1answer
27 views

How to find $\lim \sup (a_n+1)$ or $\lim \inf (a_n+1)$?

Let $\{a_n\}$ be a positive sequence. Prove $\sum a_n$ converges $\iff$ $\sum{a_n\over a_n+1}$ converges. I showed $\sum a_n$ converges $\Rightarrow$ $\sum{a_n\over a_n+1}$ converges. Proving the ...
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1answer
49 views

Show $\mathcal{H}_\eta = L^2([a,b], \eta)$ is a Hilbert space when $\eta$ is positive, not necessarily continuous

Exercise $8$ of Stein and Sharkarchi's Real Analysis asks first to show that the space of measurable $f$ on $[a,b]$ such that $$\int_a^b |f(t)|^2 \eta (t)dt < \infty $$ denotes $\mathcal{H}_\eta = ...
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2answers
43 views

Let $S$ be a nonempty subset of $\mathbb{R}$. Prove the following are equivalent?

I have no idea how to show these are equivalent. Can someone help me break this down step-by-step?
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1answer
19 views

Sign of an integral of a product function

Let $a$ and $b$ be any two real numbers and $f$, $g$ $\in$ ${C}[a, b]$. If $\int_a^b f > 0 $ and $\int_a^b g < 0$, can we say that $\int_a^b fg <0$?
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1answer
28 views

$f'$ is integrable in $[0,1]$ and $f(0)=0$ in $[0,1]$ Prove that $\ |f(x)|\le\sqrt{\int_0^1|f'|^{2}dx}$

I've tried this using the Cauchy inequality for integrals, but I can't get the algebra right, could anyone help me solving this? Thanks
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1answer
58 views

Prove that the norms ||.||2 and ||.||infinity on a finite dimensional space are equivalent [duplicate]

I have to prove that the norms $||\cdot ||_2$ and $||\cdot ||_\infty$ on a finite dimensional space are equivalent. I know how to do this with two integer values for the norms where for example ...
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2answers
80 views

Prove that lim$_{n → \infty} \int_{0}^{3} \sqrt(sin \frac{x}{n} + x + 1) dx$ exists and evaluate it

Prove that the following limits exist and evaluate them. c) $\lim_{n \to \infty} \int_{0}^{3} \sqrt{\sin \frac{x}{n} + x + 1}\ \text dx$ I need to use the following theorem from analysis; Suppose ...
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1answer
39 views

What is the $k$th differential of a composite map?

Let $V$, $W$, and $X$ be normed linear spaces, and let $A$ be an open subset of $V$. Suppose $F \in C^k(A,W)$ and $G \in C^k(W,X)$. What is $d^k(G\circ F)_\alpha$ (the $k$th differential of $G \circ ...
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0answers
67 views

difficult example of a not differentiable function $f: \mathbb R^2 \to \mathbb R^2$ [closed]

Give an example of a function $f: \mathbb R^2 \to \mathbb R^2$ so that: 1) all its directional derivatives exist at $(0,0)$ ($D_{\vec u}f(0,0)$ exist for all $\vec u \in \mathbb R^2$ unitary), 2) ...
3
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2answers
514 views

Proving a set is neither open nor closed

Prove that the set $\left\{(x,y)\in \mathbb{R}^2|0<y\le 1\right\}$ is neither closed nor open. My book only has definitions (no examples) and I understand them but have no idea how to ...
3
votes
1answer
90 views

real analysis converging proof using Abel's formula.

Suppose that $\Sigma_{k=1}^\infty a_k$ converges. Prove that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$. Attemtp: ...
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0answers
41 views

Fundamental questions about Logarithm and finding quadratic roots

Define: $(e^{iz}+e^{-iz})/2= cos z$ where $z \in \Bbb C $, i.e, the cosine function is defined for complex $z$. Now, is it true that for each $w \in \Bbb C $ there is $z \in \Bbb C $ such that $cos z ...
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2answers
35 views

A sequence of functions ${f_n}$ that converges non-uniformly to $f$ but the limit of the integrals equals the integral of the limits?

Construct an example to show that it is possible to have a sequence of continuous functions $\{f_n\}$ that converges non-uniformly to a continuous function $f$ on a closed interval $[a,b]$, but we ...
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1answer
42 views

Proof of non-integrability for Dirichlet like function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function which has the value $$f(x)=\left\{ \begin{array}{l l} x^2 & \quad ,x\in\mathbb{Q}\\ x^3 & \quad ,x\notin\mathbb{Q} \end{array} ...
1
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1answer
70 views

Rudin's theorem 6.6

The theorem states: $ f $ is Riemann integrable if and only if for $ \epsilon >0 $ there exists a partition $ P $ such that $ U(P,f)-L(P,f)<\epsilon$. ($U(P,f) $ denotes the upper sum and and ...