# Tagged Questions

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### Countable and uncountable sets.

a) Show that $\left \{ n^{2}+m^{2}:n,m\in \mathbb{N} \right \}$ is countable. b) Show that $\left \{ x\in \mathbb{R}:x(x-2)<0 \right \}$ is uncountable. My answers: a) Is it possible to define ...
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### Show that if f is strongly differentiable at $x_0$ then it satisfies Lipschitz condition in a neighbourhood of $x_0$.

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that $f$ is strongly differentiable ...
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### Show that there exists a subsequence $\{F_{n_{k}}\}$ which converges to uniformly on $[a,b]$.

Let $\{f_n\}$ be uniformly bounded sequence of functions which are Riemann-integrable functions on $[a,b]$ and define for $a\leq x\leq b$. $$F_n(x)= \int_a^x f_n(t)dt.$$ Show that there exists a ...
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### If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,…$. Prove that $f=0$ on $[0,1]$.

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,...$ $$\int_0^1 f(t)t^n dt=0$$ Then prove that $f=0$ on $[0,1]$. Here I don't want the proof. I have one proof. But I have ...
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### If a real valued function is increasing bounded above, does that imply that $\lim_{x\to \infty}f(x)$ exists?

I have a quick question: If a real valued function is increasing on $[x_0,\infty)$ and bounded above, does that imply that $\lim_{x\to \infty}f(x)$ exists ? I know that if a sequence is increasing ...
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### Which negation of the definition of a null sequence is correct?

A Cauchy sequence $a_n$ is said to be a null sequence if for every $\varepsilon>0,$ there exists an integer $N$ s.t. $\forall n >N, \lvert a_n \rvert < \varepsilon$. I thought the negation ...
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### Inequality for all real numbers

I am trying to prove a hw problem from Taos Analysis 1 book. I would like some help proving the following statements if they are true which I do not necessarily believe. Let $x,y \in \Bbb R$. Show ...
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### (absolute) Convergence of a series

I want to prove that the following series is convergent for $x>0$: $$\sum_{n=1}^\infty \left( \prod_{p\mid n} \frac{1}{p-1}\right) n^{-x}$$ I tried to estimate the product but I didn't get so ...
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### $C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
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### If $f$ is equal to an affine function up to $1$-th order at $a$, then $f$ is differentiable at $a$, proof more subtle then it appears?

I came across the following exercise: Two functions $f, g : \mathbb R \to \mathbb R$ are equal up to $n$th order at $a$ if $$\lim_{h \to 0} \frac{f(a + h) - g(a + h)}{h^n} = 0.$$ Show that $f$ ...
Suppose that $$f(x)=\sum_{n=1}^{\infty}f_n(x)\,\,\,\,\,\,\,\,\,\,(x\in X)$$Where $f_n:X\rightarrow [0,\infty]$ for $n=1,2,3...$ Let $g_N=f_1+...+f_N$. Then the sequence $\{g_N\}$ converges ...