Tagged Questions

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Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
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Topology, Proof of function being continuous

Let $(X_i,d_i),(Y_i,d_i^*)$, $i=1,\ldots,n$ be metric spaces. Let $f_i:X_i \to Y_i, i=1,...,n$ be continuous functions. Let $$X = \prod_{i=1}^{n} X_i , Y = \prod_{i=1}^{n} Y_i$$ and ...
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Proving IMVT using delta-epsilon

Let's assume $f(a)<0$ and $f(b)>0$. IMVT claims that there's $c\in(a,b)$ such that $f(c)=0$. The Proof: Consider $$A = \{ a\le x\le b : f(x) < 0 \}$$ That's a non-empty set and therefore, by ...
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$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
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If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
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$\{f_n\}$ equicontinuous sequence of functions on compact $K$, converges pointwise on $K$ then converges uniformly on $K$.

Suppose $\{f_n\}$ is an equicontinuous sequence of functions on a compact set $K$ and $\{f_n\}_{n=1}^{\infty}$ converges pointwise on $K$. Prove that $\{f_n\}_{n=1}^{\infty}$ converges uniformly on ...
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Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
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Prove $mx+b$ is continuous at any point in $\mathbb{R}$

I need to prove, $mx+b$ is continuous at any point in $\mathbb{R}$ Now, as I have thought of there's 2 possible cases: 1) $m = 0$ 2) $m \neq 0$ So for case #2, $m < 0 \vee m > 0$ , and we ...
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Verification of $\epsilon$-$\delta_\epsilon$ definition of continuity proof

Show that the function $$f(x) = \begin{cases} \frac{1}{q}, & \text{if x = \pm \frac{p}{q} in the lowest terms with p, q \in N} \\ 0, & \text{if x \in R - Q} \end{cases}$$ ...
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Introduction to Analysis: Locally and Actually Constant

I was given this problem for homework. I more a less understand it. I just need to somehow finalize my ideas. The problem reads: Prove that a function which is locally constant on $[0,1)$ is ...
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Introduction to Analysis: Continuity and Limits

My coworker and I were looking at a problem for our Real Analysis class. It reads: Call a function "multiplicatively periodic" if there is a positive number c $\neq$ 1 such that $f(cx) = f(x)$ for ...
Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.
Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...