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.

learn more… | top users | synonyms

0
votes
0answers
6 views

Prove Property of Doubling Measure on $\mathbb{R}$

Problem. Let $\mu$ be a fixed finite measure on $\mathbb{R}$. $\mu$ is said to be doubling if there exists a constant $C>0$ such that for any two adjacent intervals $I=[x-h,x]$ and ...
0
votes
0answers
26 views

Imaginary part of $f_m$ tends to zero

Does anybody have an idea how to show that for $|x|< \pi$ the imaginary part of the following sequence of functions $f_m$ tends to zero for $m \rightarrow \infty.$ $$f_m(x):=\left( ...
5
votes
0answers
18 views

Evaluating $\int_0^1 \frac{z \log ^2\left(\sqrt{z^2+1}-1\right)}{\sqrt{1-z^2}} \, dz$

What real analysis tools would you employ for this kind of integral? $$\int_0^1 \frac{z \log ^2\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$
2
votes
1answer
41 views

Misunderstanding of $\epsilon$-neighborhood

I am given the following definition of an $\epsilon$-neighborhood: Given a real number $a\in \mathbb{R}$ and a positive number $\epsilon>0$, the set $$\{x \in \mathbb{R}: |x-a|<\epsilon \}$$ ...
3
votes
3answers
44 views

Prove sequence $\left(\frac{1}{6n^2+1}\right)$ converges to $0$

I am asked to verify that the sequence $\left(\frac{1}{6n^2+1}\right)$ converges to $0$: $$\lim \frac{1}{6n^2+1}=0.$$ Here is my work: $$\left|\frac{1}{6n^2+1}-0\right|<\epsilon$$ ...
2
votes
2answers
50 views

Mean Value Theorem and Inequality.

Using the mean value theorem prove the below inequality. $$\frac{1}{2\sqrt{x}} (x-1)<\sqrt{x}-1<\frac{1}{2}(x-1)$$ for $x > 1$. I don't understand how these inequalities are related. Am I ...
0
votes
1answer
21 views

Help understanding definition of Darboux integral $U(f)$.

My book defines the upper and lower Darboux sums $U(f,P)$ and $L(f,P)$ respectively then follows up with a confusing definition of the upper and lower Darboux integrals $U(f)$ and $L(f)$ respectively. ...
2
votes
0answers
35 views

Unique extendable functions… Is there a theory?

Motivation: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function from $X$ to $Y$ that coincides with $f$ in a dense set, then $g=f$. ...
1
vote
2answers
40 views

How to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$

I want to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$ if $p > 1$ and $r \in \mathbb{N}$. To show this, I wanted to use that $\lim_{x \to 0} x \ln x = 0$, and in fact if $p \geq r$ we can write ...
0
votes
0answers
22 views

Show that the solution of this differential equation is analytic

Let $\alpha,\beta,a,b$ be real constants. Show that the differential equation given by: $y''= ay' + by \\ y(0)=\alpha\\ y'(0)=\beta$ has just one solution which is analytic in $R$ Solution: I am ...
5
votes
0answers
45 views

Disprove, fix, prove: If {$a_n$} and {$b_n$} are increasing, then {$a_n b_n$} is increasing

Prove the statement wrong, fix it, then prove the new statement: If {$a_n$} and {$b_n$} are increasing, then {$a_n b_n$} is increasing I think I'm headed in the right direction with this but I'm ...
-2
votes
0answers
20 views

Convergence of the geometric mean of a converging sequence [on hold]

Show that if $(x_n)$ is a convergent sequence, then the sequence given by $y_n:=(x_1\cdot x_2\cdot x_3\cdot \ldots\cdot x_n)^{\frac{1}{n}}$ converges to the same limit.
0
votes
1answer
22 views

Compactly supported functions; continuous continuation?

While I was trying to prove something else, I stumbled over this question. Let $f \in C^2_c(\mathbb{R}^n; \mathbb{R})$ be non-negative everywhere does this mean that $$g(x):=\frac{||\nabla ...
2
votes
1answer
38 views

Definition of the reals and basic proofs are independent of the axiom of choice? [duplicate]

I default to the construction of the reals I learned in my first semester Real Analysis course. This is that they are the set of equivalence classes of convergent Cauchy sequences of rationals. I am ...
2
votes
2answers
32 views

Prove bounds of a strictly increasing sequence using integrals to approximate

For the strictly increasing sequence $ \ x_n = \frac1{1^2} + \frac1{2^2} + \frac1{3^2} +\cdots+\frac1{n^2},$ for $n\ge1$. (a) Prove the sequence is bounded above by $2$; deduce that is has a limit ...
4
votes
1answer
37 views

A question about a polynomial

Suppose that $p$ is a real polynomial of degree $n$. Prove that for $|x|<1$, $$\sum\limits_{m=0}^\infty{p(m)x^m}=h((1-x)^{-1})$$ for some real polynomial $h$ of degree $n+1$ without the ...
2
votes
0answers
19 views

Limits of integral ratio: which theorems to use?

I "guessed" the limit $$ \lim_{n\rightarrow+\infty} \frac{\int_{a}^{b} f(x,y)\cdot g(x,y)^ndy}{\int_{a}^{b} h(x,y)\cdot g(x,y)^ndy}=\frac{f(x,a)}{h(x,a)}.$$ where $f,h>0$ and $f,g,h\in ...
-1
votes
1answer
28 views

show that $f$ is concave iff $-f$ is convex.

let $f : S \subset \Bbb R^n \to \Bbb R$ be a function. then, I want to show that $f$ is concave iff $-f$ is convex. definition of convexity: $x,y\in S$ and $\alpha \in (0,1)$ $f(\alpha ...
0
votes
1answer
25 views

Continuity in closed sets

Please help me, I have being trying this for days now. Let $f:F \to \mathbb{R}$ be a function on a closed set $F$. Show that $f$ is continuous if and only if $A=\{x \in F; f (x) \leq c\}$ and $B=\{x ...
0
votes
0answers
24 views

Taking Analysis I, Abstract Algebra I, and Theoretical Linear Algebra [on hold]

S.E advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the ...
5
votes
3answers
60 views

Wave equation $u_{xx}+u_{xt}- u_{tt}=0$

Does anybody know how we can solve the equation $u_{xx}+ u_{xt}- u_{tt}=0$ with $u(x,0):=g(x)$ and $u_t(x,0):=h(x)?$ I mean it is known how to do this for the wave equation see here but I don't know ...
1
vote
1answer
18 views

Converging subsequences and subsets having infinite elements

We have a metric space (V,d). Proof that the following two properties are equivalent. a) Every sequence $a_n \in V$ has a subsequence which converges to a element $x \in V$ b) For every ...
2
votes
1answer
33 views

Circular definition of tangent line and derivative

I'm trying to understand the deep relations between the tangent line to the graph of a function $f$ at a given point $P$, and the derivative of $f$ at the same point. Indeed, in many books the ...
2
votes
3answers
38 views

On an Integral inequality.

I am following a proof and I am having troubles with the last inequality stated Specifically could I have some extra passages on this? $$\int_{\delta}^{\pi} [f(w+u) - f(w)] \frac{\sin^2(nu/2)}{2 ...
6
votes
1answer
63 views

Bound on first derivative $\max \left(\frac{|f'(x)|^2}{f(x)} \right) \le 2 \max |f''(x)|$

I want to show that for a function $f \in C_c^2((a,b))$ non-negative, the inequality $$\sup \left(\frac{|f'(x)|^2}{f(x)} \right) \le 2 \sup |f''(x)|$$ holds. I noticed that the left term is equal ...
1
vote
1answer
32 views

On improper Riemann integral

Let $F:R \to (0,1)$ be a strictly increasing continuous distribution function such that $\int_{-\infty}^{\infty}xdF(x)<\infty$. Let $(x_k)_{k \in N}$ be a sequence of uniformly distributed numbers ...
2
votes
1answer
19 views

Prob. 14, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hermitian positive semi-definite form

Let $X$ be a complex vector space, and let the map $h \colon X \times X \to \mathbb{C}$ satisfy the following conditions: For all $x, y, z \in X$ and $\alpha \in \mathbb{C}$, (i) $h(x+y, z) = ...
4
votes
0answers
71 views

Is there a math basis for what I observe in simulations? [duplicate]

I am stuck in my research with the following problem which occurs when proving a theorem. If $x,y\in \mathbb{R}^+$ and $x>y$, is it true that $(x-y)^{((x-y)/(2x-y))}\times ...
2
votes
0answers
19 views

Prove boundedness of 2nd derivatives

Let $f \, \colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a smooth and convex function. Assume $f$ behaves asymptotically as a cone at infinity, i.e., $ \lim_{R \rightarrow \infty} \frac{f(R x)}{R} ...
-1
votes
0answers
22 views

A Hilbert space is isomorphic to its second dual?

How to show that any Hilbert space $H$ is isomorphic to its second dual space $H^{\prime\prime} = (H^\prime)^\prime$? (This is Prob.8, Sec. 3.8 in Erwine Kreyszig's Introductory Functional Analysis ...
-3
votes
0answers
61 views

Proving using AM-GM inequality

If $x,y\in \mathbb{R}$, and $x>y$, how to show $(x-y)^{((x-y)/(2x-y))}\times (x+y)^{((x)/(2x-y))}>x$? I know I have to use AM-GM inequality, but it is not clear how.
0
votes
1answer
27 views

Proving that a solution exists

Proof that there exists a $x>0$ with $x \in \mathbb{R}$ s.t. $\sin(x) = \frac{x}{2}$ I tried to use the intermediate value theorem, but I don't know how to apply it correctly. Obviously ...
0
votes
1answer
18 views

Sum of elements in a sequence

Let $a_n$ be a sequence in $\mathbb{R}$ and $a\in\mathbb{R}$. Suppose that $N \in \mathbb{N}$, $\epsilon >0$ and for every $n > N$ $|a_n -a|<\epsilon$. Show that for every $n>N$ the ...
1
vote
5answers
108 views

Is $\sqrt{2t \log{\log{\frac{1}{t}}}}$ increasing for $t\in(0,a)$, for a suitable $a>0$?

I believe it's obviously, but I tried a lot, and have no clue how can I show that $$\sqrt{2t \log{\log{\frac{1}{t}}}}$$ is increasing for $t\in(0,a)$ for a suitable $a>0$?
0
votes
1answer
48 views

Directional derivative $f(x,y)=\frac{x^3}{1+x^2+y^2}$

I'm stuck on calculating the directional derivative of $f(x,y)=\frac{x^3}{1+x^2+y^2}$ in $(3,-1)$ along $(a,b)\in\mathbb{R}^2$. My try: $\lim\limits_{t\to ...
5
votes
0answers
38 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
2
votes
1answer
32 views

Simple points of an algebraic variety from an analytic point of view

I am a specialist in fuctional analysis, but from time to time I have to use some results from algebraic geometry, and every time I face great difficulties in translating them into the language ...
3
votes
1answer
43 views

Is there a total summation function?

I define a summation function to be a partial function $F$ from infinite sequences of real numbers to the extended reals, such that: (1) Sequences which are zero in all but possibly one position are ...
-2
votes
0answers
31 views

How can I resolve: $ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $ via numerical solution?

How can I resolve a second-order ODE via Euler method? By example in the next ODE: $$ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $$ I know Euler method: $x_{i+1} = x_{i} + ...
2
votes
1answer
28 views

Showing an integral is in $L^1$

Let $0<a<1$ and $f\in L^1([0,1])$. Show $g(x)=\int_0 ^x\frac{1}{(x-t)^a}f(t)dt$ exists a.e. in $[0,1]$ and $g\in L^1([0,1])$. Using Fubini, $$\int_0 ^1 \vert g(x) \vert dx=\int_0 ^1 \int_0 ...
1
vote
1answer
29 views

Prob. 7, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of a Hilbert space is a Hilbert space.

Here's Prob. 7, Sec. 3.8 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that the dual space $H^\prime$ of a Hilbert space $H$ is a Hilbert space with inner product ...
-1
votes
1answer
18 views

Continuous function and its Mapping

Let $f: \mathbb R \to \mathbb R$ be a continuous function. Which one of the following sets cannot be the image of $(0,1]$ under $f$? $\{0\}$ $(0,1)$ $[0,1)$ $[0,1]$. We know that $(0,1]$ is ...
3
votes
2answers
71 views

Prove this limit $\lim \limits_{x\to\infty}f(x)=0$

I have this problem in real analysis. I think it needs integral factor or knowledge of ODE to prove, but not sure how to it. Here is the question: Let $f$ be a real valued continuous function on ...
1
vote
1answer
16 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
1
vote
0answers
17 views

$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n)$

I am trying to prove the equality $$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n),$$where $\mathcal B(\mathbb R^i)$ is the Borel $\sigma$-algebra on $\mathbb ...
3
votes
2answers
43 views

Question on proof of Heine-Borel theorem

Spivak's book on calculus on manifolds has a statement that I can't grasp. Say we have the closed interval $[a,b]\subset\mathbb{R}$ covered by $\mathcal{O}$ and we define $$A=\{x \in [a,b]:[a,x] ...
1
vote
0answers
14 views

inverse function theorem for analytic functions whose derivative might vanish

Suppose $x(t), y(t)$ are monotone increasing functions, and $f$ and $g$ are real-analytic functions that are not identically zero. If $f(x(t)) = g(y(t))$ for all $t$, does it follow that $x$ is an ...
0
votes
1answer
32 views

if $\lim a_n = \infty$ and $\lim b_n = B$, then $\lim (a_n+b_n) = \infty$

I'm having trouble starting the proof not sure exactly how to go about it. So far I know that for a sequence to go to infinity it means that for all $n >0$ there exists $n_0$ for all $n$ greater ...
3
votes
1answer
48 views

A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
0
votes
0answers
25 views

An interval covering problem [on hold]

Consider a set $A$ of intervals $[a_i,b_i)$, whose union is $[0,1)$. Prove there must exist a subset $B$ of $A$ such that the intervals in $B$ are pairwise non-overlapped and the sum of their lengths ...