# Tagged Questions

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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### Taking limits on each term in inequality invalid?

So this inequality came up in a proof I was going through. $$c - 1/n < f(x_n) \leq c$$ Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a ...
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### Non-Cauchy products for series?

Let $\sum a_n$ and $\sum b_n$ be two absolutely convergent series. My question is Is it possible to take the product of the two series $(\sum a_n)(\sum b_n)$ and get a result different from the ...
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### Is it true that $\sin x > \frac x{\sqrt {x^2+1}} , \forall x \in (0, \frac {\pi}2)$?

Is it true that $$\sin x > \dfrac x{\sqrt {x^2+1}} , \forall x \in \left(0, \dfrac {\pi}2\right)$$ (I tried differentiating , but it's not coming , please help)
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### Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
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### Formal analysis proof for specific limit $\large{|x - \frac{p}{q}| < \frac{1}{n}}$ [closed]

If $x \in \mathbb{R}$ and $n \in \mathbb{N}$ then there exists $p, q \in \mathbb{Z}$ such that $$\left|x − \frac{p}{q}\right| < \frac{1}{n}.$$ Do we use the Archimedian Principle to prove ...
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### Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
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### Every function $f: \mathbb{N} \to \mathbb{R}$ is continuous?

This is a question that came up as a true false question in my textbook, and I was wondering what you thought of my reasoning. I claim that even though a graph of such a function doesn't look ...
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### The boundary of the union of two sets is a subset of the union of boundaries

I'm stuck on trying to get this proof started. I want to prove that $\delta(S_1 \cup S_2)\subset \delta S_1\cup\delta S_2$, where $S$ is some set. I don't need a full proof, just a hint to get ...
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### how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
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### Help understanding proof on Jensen's Inequality

I need help understanding the proof for Jensen's inequality in "Real and Complex Analysis" by Rudin. 3.3 Theorem (Jensen's Inequality) Let $\mu$ be a positive measure on a $\sigma$-algebra ...
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### Sum of quotients

Assume $0<x_i\leq y<z$ for $i=1\ldots,n$. Is there an easy argument to show $$\frac{x_1}{y}+\sum_{i=1}^{n-1} \frac{x_{i+1}}{x_i}+\frac{z}{x_n}\geq n+\frac{z}{y}?$$ For $n=1$ the statement is ...
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### Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to ...
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### How to prove that E's limit point must be in E? (rudin)

How to prove that E's limit point must be in E? Thm 2.23 E's open iff E_c is closed. First, suppose E_c is closed. Choose x belongs to E. Then x doesn't belongs to E_c, and x is not a limit point ...
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### Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
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### Function that decays faster than any polynomial, but not in the Schwartz space?

Motivated by the very restrictive condition imposed in the definition of the Schwartz space, I was wondering about the following question. Is there a $C^\infty$ function that decays faster than ...
### Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.
This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
### Calculate roots from $\frac{x \cosh(x) - \sinh(x)}{x^2}$
I want to solve the following equation $$f(x) = \frac{x \cosh(x) - \sinh(x)}{x^2} = 0$$ Because the term above is undefined for $x = 0$ I calcuted \lim_{x \rightarrow 0}\frac{x \cosh(x) - ...