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, integration through the fundamental theorem of calculus.

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18 views

Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
2
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1answer
20 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
1
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1answer
35 views

Conceptual question about Riemann Integral

I've been working on some problems involving Riemann integrals, and I been having problems relating $sup\{\mathcal{L}(\mathcal{P},f)\}$ to some other known quantity, like $\mathcal{L}(\mathcal{P},f)$ ...
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1answer
34 views

Example of $\liminf_{n\rightarrow\infty}(x_n+y_n)>\liminf_{n\rightarrow\infty}x_n+\liminf_{n\rightarrow\infty}y_n$

A result from introductory analysis shows that given two bounded sequences $\{x_n\}$ and $\{y_n\}$, ...
1
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1answer
31 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
0
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1answer
24 views

Prove that the limit exists of an increasing and bounded function

This was an exam question I had last year but wasn't able to answer it (and still can't). Suppose $a<b$, and $f : (a,b) \to \mathbb{R}$ is a function that is both increasing and bounded. Prove ...
3
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3answers
72 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
0
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1answer
39 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
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0answers
29 views
+50

Question concerning the integrability of a function

Let $f: [0,1]^2 \to \mathbb{R}$ be a function such that $$ f(x,y) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q} \\ 2y & : x \notin \mathbb{Q} \end{array} ...
2
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3answers
47 views

If $f\in L^1$ has a compact support and $0 \leq p \leq1$ then $|f|^p\in L^1$

My text proved that If $f\in L^1$ is bounded and $p \geq1$ then $|f|^p\in L^1$ I wanted to prove the seemingly very similar statement: If $f\in L^1$ has a compact support and $0 \leq p ...
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0answers
23 views

What is the value of those limits?

$$\lim_{n\longrightarrow{\infty}}{\int_{0}^{\infty}{\arctan{(nx)}e^{-x^n}dx}}$$ And $$\lim_{n\longrightarrow{\infty}}{\int_{0}^{+\infty}{(1+\frac{e^{-nx}}{\sqrt{x}})(1-\tanh{(x^n)})dx}}$$
2
votes
1answer
48 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
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1answer
16 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
1
vote
1answer
29 views

Show that any infinite set $X$ may be endowed by a metric d such that $X$ has a limit point in $(X,d)$

This is an exercise I've been dealing with for a few days; I was wondering if anyone could help me with a hint or just telling me the answer. Regards
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1answer
22 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
2
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2answers
27 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
1
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1answer
32 views

Does bounded variation imply boundedness

Using the standard definition $$||f||_{TV} := \sup_{x_0<\cdots<x_n}\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$$ 1.When the domain is a bounded interval $[a,b]$, the statement holds. 2.When the ...
7
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1answer
105 views

Calculate the following Integral (Please Help)

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
0
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1answer
28 views

The Simplex in $\mathbb{R}^n$ is convex

Problem: Show that $S:= \lbrace v \in \mathbb{R}^m \mid v=\displaystyle \sum_{j=1}^n a_j v_j, \text{ with } a_1, \dots , a_m \in [0,1], \ \sum_{j=1}^m a_j=1 \rbrace$ the Simplex of $\mathbb{R}^n$ ...
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0answers
54 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
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3answers
39 views

Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$

For $a_n \geq 0$, and $\sum a_n$ convergent, show that $\sum \frac{\sqrt{a_n}}{n^p}$ is also convergent for $p > 1/2$? What bugs me more is why isn't $\sum \sqrt{\frac{a_n}{n}}$ convergent?? ...
0
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1answer
16 views

Finding the “most” continuous representative of a class of functions equal almost everywhere.

In measure theory, we consider functions to be basically the same if they are equal almost everywhere. It seems crazy, though, to choose any of these as the representative when doing calculations. Why ...
2
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2answers
31 views

$\Lambda$ for which $w(k)= k^3+8\Lambda \sin(k)$ has a minimum on x-axis.

Given equation $$\begin{array}{cc} \omega(k)= k^3+8\Lambda \sin(k) & \textrm{ with } k>0, \Lambda>0 \end{array}$$ Clearly $\omega(k)$ has a minimum for $k \approx 4$ which i will call ...
1
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2answers
29 views

Infinite series of a function involving an enumeration of rationals on $0,1$

Let $\{ q_n : n \in \mathbb{N} \}$ be an enumeration of the rational numbers in $(0,1)$ and define $f_n(x) = \begin{cases} 0 \qquad \text{if} \; x \in (0, q_n) \\ 2^{-n} \quad \: \text{if} \; x \in ...
-1
votes
1answer
20 views

an example of almost uniformly convergence [on hold]

$x^n\rightarrow 0 $ on [0,1]. does $x^n$ converge almost uniformly to zero. if we take out point 1 or if take out a small set $[1-\epsilon,1]$ why does not it a.u. converge when we take only point ...
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2answers
31 views

Find convergence or divergence of Series.

Determine which of the following series converge. Justify your answers. a) sum of (sin$(\frac{n \pi}{6}))^n$ b) sum of (sin$(\frac{n \pi}{7}))^n$ I believe that both sequences diverge because a sin ...
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2answers
37 views

Let $f(z):=e^{\frac 1 z}, {z \in \mathbb C \setminus\{0\}}$.What values of $z$ are $ f(z)=re^{i\phi}$ for $r\in(0, \infty), \phi\in\mathbb R$?

Consider $f(z):=e^{\frac 1 z}, {z \in \mathbb C \setminus \{0\}}$. For which values are $f(z)$ real ? I've considered $e^{\frac 1 {a+ib}} = e^{\frac {a-ib} {a^2+b^2}}$. For which values are $f(z)$ ...
2
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1answer
47 views

Is this just asking for the Uniqueness of Limits? If not, how can you do this?

Suppose real function $f$ is continuous at every point. Prove that $f(x)=c$ cannot have two solutions for every value of $c$. I think you can prove this by contradiction assuming there are two ...
0
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1answer
32 views

measurable subset of nonmeasurable set

show that if E is measurable and E⊂P where P is nonmeasurable set in [0,1), then m(E)=0. Can one please tell how to start .. and I have one more question: is the union of m'ble set and non-m'ble set ...
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0answers
50 views

Why f' is zero a.e. where f zero?

We have function $f:\mathbb{R}\rightarrow \mathbb{R}$. How to proove that $f'(x)=0$ a.e. on $\left\{x | f(x)=0 \ \mbox{and}\ \exists f'\right\}$?
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1answer
15 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
0
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1answer
21 views

Sufficient condition for equality of two radon measures

Let $ X $ be a locally compact Hausdorff space and let $ \phi_1 $ and $ \phi_2 $ be two Radon measures on X (outer measure means measure and the definition of Radon measure that I am assuming can be ...
0
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1answer
27 views

How would Rolle's Theorem be used to prove this?

g is differentiable on (a,b) and for all x in [a,b] $a<g(x)<b$, and $|g'(x)|<1/2$. Prove $g(x)=x$ has at most one solution in [a,b]
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1answer
29 views

$\epsilon - N$ proof confirmation.

These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it. My limit and required assumptions: ...
0
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1answer
42 views

Why an integral does not exist?

I am trying to construct a counter example of Fubini Thorem, and for that we need a function $f$ in the product space which is not absolute integrable. So, let ...
3
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1answer
34 views

Showing $\lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4}$

How could I go about proving the following limit: $$ \lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4} $$
0
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1answer
28 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
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0answers
12 views

Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
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2answers
131 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
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1answer
44 views

Real Analysis question on sequences (Hint needed!!!)

Given the number $\alpha > 1$ , define the sequence an where $a_0 = 1$ and $a_{n+1} = (\alpha \times a_{n})^{\frac{1}{4}}$ for $ n \geq 0 $. Prove: If $a_{n}^{3}< \alpha $(as is true when n ...
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0answers
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+200

Take 2: When/Why are these equal?

This didn't go right the first time, so I'm going to drastically rephrase the query. As per this previous question, I am wondering if the two series ...
2
votes
1answer
59 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
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1answer
39 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
1
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1answer
15 views

Proving that sng(x) is discontinuous at 0?

I am trying to prove that the signum function is discontinuous at $x_0 = 0$. The criterion for discontinuity is that if there is a sequence $(x_n) \subseteq A$ and $(x_n) \rightarrow c$ but $f(x_n)$ ...
3
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1answer
35 views

When are these series equal?

Suppose we have a power series $$\sum_{n=0}^\infty {a_nb_nx^n}$$ When is it true that the series obtained by eliminating $b_n$ is proportional to the original series? $$\sum_{n=0}^\infty ...
5
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2answers
130 views

Subsets of $[0,1]$

Suppose we have a closed subset $A\subset[0,1]$ that is not equal to $[0,1]$. Is it possible $mA=1$? Suppose you have an open subset $B\subset[0,1]$ that is dense in $[0,1]$. Is it possible that ...
0
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3answers
41 views

How do you evaluate $f(x_n)$?

I am not sure I am clear on how to evaluate the $f(x_n)$; what the it mean to evaluate the function of a sequence?
1
vote
1answer
41 views

Epsilon Delta Proof of a limit [on hold]

How to write a formal epsilon delta proof for the following limit, $$\lim_{x \to 1} \frac{x^2+x-2}{x-1}$$
2
votes
1answer
30 views

$(\epsilon,N)$ proof that $\displaystyle{\lim_{n \to \infty} \frac{n^3-3n^2+4n-7}{2n^3-10n^2+1} = \frac{1}{2}}$

I need to find an $N \in \mathbb{N}$ such that: for all $\epsilon>0$, there exists $N>0$ such that for all $n \geq N$ $$\left|\frac{n^3-3n^2+4n-7}{2n^3-10n^2+1} - \frac{1}{2} \right| < ...
1
vote
1answer
46 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!