# Tagged Questions

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### Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
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### Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$\int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
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### If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N.$$ Suppose there exists a continuous function $g$ on ...
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### Checking proof that $f(x)=x^2+1$ is continuous

Let $f:\mathbb R \to \mathbb R$ is defined by $f(x)=x^2+1$. Prove this function is continuous for all $x \in\mathbb R$. Here is what I have: Suppose that $c∈ℝ$. Let $\varepsilon>0$. Let ...
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### Operator $Au(t) = \int_0^t e^{t-s} u(s) ds$ (Proof Verification)

Consider the space $C([0,1])$ with $||\cdot||_\infty$ norm. Let $A: C([0,1])\rightarrow C([0,1])$ be the operator defined by $$Au(t) = \int_0^t e^{t-s} u(s) ds.$$ And I am not 100% sure about (c), ...
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### Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
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### Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...
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### “Simple” question (Lebesgue Integration) in a hard exam (proof verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
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### How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
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### $B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
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### Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
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### Characteristic function of Cantor set is Riemann integrable

I want to prove that the characteristic function of the Cantor set is Riemann integrable on $[0,1]$. Could somebody please tell me if my proof is correct? Let $f$ be the characteristic function of ...
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### Convergence of measures — revisited

In this thread, I asked a question about the convergence of measures. The conjecture I posed there, which turned out to be false, was supposed to be a lemma that I wanted to use to prove a ...
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### Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
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### Proof Verification: $(ab)^{-1}=a^{-1}b^{-1}$

I am asked to prove $$(ab)^{-1} = a^{-1}b^{-1}$$ where $a,b\in\mathbb{R}\setminus\left\{0\right\}$. Here is what I have: $$(ab)^{-1} = \frac{1}{ab} = \frac{1}{a} * \frac{1}{b} =a^{-1}b^{-1}$$ ...
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### Proving the Nested Interval Property using Axiom of Completeness

I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem ...
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### Changing one point does not change the Riemann integral

I tried to prove the following. Please could somebody tell me if my proof is correct? Let $f: [a,b]\to \mathbb R$ be Riemann integrable. Then changing one value of $f$ then $f$ is still ...
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### Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
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### Alternative definition of complex number, showing it is equivalent to the tradidional one.

The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$. Here is his ...
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### Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist

Hello I was hoping that someone might be able to verify that the following proof that $\lim_{x\to 0} {1\over x}$ does not exist is correct. First assume that $\lim_{x\to 0} {1\over x} = L$. This ...
### Suppose $a_n \geq 0$, and $\sum a_n$ diverges, and $\lim a_n = 0$. Show that $\sum \frac{a_n}{1+a_n}$ diverges.
Can someone please verify my proof sketch? Suppose $a_n \geq 0$, and $\sum a_n$ diverges, and $\lim a_n = 0$. Show that $\displaystyle{\sum \frac{a_n}{1+a_n}}$ diverges. Let $\epsilon > 0$. ...