# Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

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### Arzelá-Ascoli Theorem precompact sets

Is the Arzelá-Ascoli Theorem true in a precompact subset of $\mathbb{R}^n$?. If $S \subset \mathbb{R}^n$ is precompact and we have a sequence $(f_n)$ of functions in $C(S)$ (Space of bounded and ...
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### Prove that $C_p[-\pi,\pi]$ is complete, knowing that $C[-\pi,\pi]$ is indeed complete

Show that the space $$C_p[-\pi,\pi] = \{f \in C[-\pi,\pi] \; | \; f(-\pi)=f(\pi) \}$$ is complete knowing that the space $C[-\pi,\pi]$, i.e. continuous functions on $[-\pi,\pi]$, is complete. Here is ...
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### Convergence in measure iff every subsequence has a convergant subsequence

My task is to show that on a finite measure space $(X,A)$, if $f_n \to f$ in measure, then every subsequence of $\{f_n\}$ has a subsequence that converges to $f$ almost everywhere. I believe I was ...
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### Sum of an odd recursive sequence

Let $a_0 = 1$ $a_1 = 1 - \frac{e}{2}$ $a_n = \frac{e}{2^n} - \frac{1 - a_{n-1}}{n - 1}$ for $n > 1$. Find $\sum_{r=0}^{\infty}a_r$.
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### Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
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### Pointwise and uniform convergence of a piecewise sequence of functions on the closed, punctured disk, $\overline{D}\prime(0,1)$.

Consider the sequence of functions $$f_n(z) = \begin{cases} n, & \text{if 0<|z|\leq\frac{1}{n}} \\ \frac{1}{z^4}, & \text{if \frac{1}{n}<|z|\leq1} \end{cases}$$ for $n\geq 1$, on ...
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### Uniform Continuity implies boundedness in sequence of functions

We want to show that, letting $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions on $[0,1]$ that's uniformly convergent, then there exists $M>0$ such that $\forall n\in\mathbb{N}$, $|f_n(x)|\leq M$...
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### Finding sequence of functions with compact support for Integral of given function

Let $a>0$ and $f(x):\mathbb{R}\rightarrow\mathbb{R}$ with $$f(x):\begin{cases}\frac{1}{\sqrt{a^2-x^2}},& x\in(-a,a)\\ 0,&\text{else} \end{cases}$$ I now have to construct sequences of ...
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### Can i prove the limit of a function using the definition of sequence limits?

So I'm asked to prove the $\lim_{x\to 1} \frac{4x+3}{x^2+4x-3}$ and of course it can be easily found $\frac{7}{2}$. However, I know I can prove it with epsilon delta definition but frankly that's ...
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I have a integral with the following property: $$\forall n>0,\quad \int_X f_n(x) dx> 0$$ I am trying to be able to say that: $$\int_X \hspace{.5em}\lim_{n\rightarrow 0} \hspace{.5em}f_n(x)dx ... 1answer 108 views ### Prove endpoints of Convergence Interval of Power Series are divergent Part a) of a question required showing that the radius of convergence of the power series \sum_{} \frac{n!}{n^n}x^n and \sum_{} \frac{n^n}{n!}x^n are e and 1/e respectively. This was fairly simple.... 1answer 40 views ### Continuous mapping theorem for a sequence of densities? Let {f_n(x)} be a sequence of densities that uniformly converges to f(x) almost surely, that is,$$ f_n(x) \xrightarrow[]{\text{a.s.}} f(x), \quad \text{uniformly},$$or equivalently$$ \Pr\...
Compute the following limit: $\lim\limits_{n\to \infty}2n \int \limits_0^1\dfrac{x^{n-1}}{1+x}\,dx.$ The value is 1. I have done this by squeezing it. Is there any other way to evaluate this ?