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.

248 questions
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Is this entrance exam question correct regarding uniform convergence?

Q. Consider the sequence of real-valued functions $\{f_n\}$ defined by $$f_n(x)=\frac {1}{1+nx^2}.$$ Assuming the fact that $\{f_n\}$ converges uniformly to a function $f$ find out real numbers $x$ ...
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Calculate $\displaystyle\lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx$.

I need calculate $\displaystyle\lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx$. I showed that $f_n(x)=\frac{nx}{1+n^2 x^3}$ converges pointwise to $f\equiv 0$ but does not converge uniformly in [0,...
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$F$ normal in $H(\Omega) \iff$ normal in $U$

Prove $F$ is normal in $H(\Omega) \iff$ for every $z\in H(\Omega)$ there is a neighborhood $U$ of $z$ such that $F$ is normal in $U$. The direct implication is immediate because $U\subset \Omega$. ...
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Relation between uniform convergence and derivatives

I know the following relation between uniform convergence and derivatives: If $(f_n)$ is a sequence of differentiable functions on $[a,b]$ such that $\lim_{n\to\infty} f_n(x_0)$ exists (and 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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Let $X$ be a metric space with metric $d$.Then $d:X \times X\rightarrow \mathbb R$ is continuous?

Let $X$ be a metric space with metric $d$.Then $d:X \times X\rightarrow \mathbb R$ is continuous? Please Check the following proof- We'll try to show this via sequential crieria of continuity....
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Show that $f_n (x)$ converges to $f(x)=0 \ \forall x$

$f_n$ is defined on $[0,1] \ \ \forall n \in \mathbb{N}$, $f_n(x)=\begin{cases} n(1-nx) &\mbox{if } \ 0 \leq x < \frac{1}{n} \\ 0 & \mbox{if } \ \frac{1}{n} \leq x \leq 1 \end{cases}$ ...
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Continuity of limit on a compact set implies uniform convergence [duplicate]

Suppose $f_n$ is a sequence of increasing functions defined on $[a,b] \to \mathbb{R}$. Say that $f_n \to f$ pointwise, such that $f$ is continuous. Does this imply that the convergence is also ...
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Suppose $f_n : [0,1]\rightarrow\mathbb{R}$ is a sequence of $C^1$ functions that converges pointwise to $f$.

I came across this problem in Davidson's real analysis text and wanted some input on whether my thinking is valid. This is section 8.1, problem H. Here is the full statement of the problem. Suppose ...
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Sequence of uniform convergent sequence $\{f_n\}$ such that $\{f_n'\}$ does not converge [closed]

I would like to find uniformly convergent sequence of differentiable functions $f_n:[0,1]\to\mathbb{R}$ such that the sequence $f_1'$, $f_2'$,$f_3'\ldots$ does not converge.
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Prove that the limit of a sequence of polynomials a polynomial

Suppose that $(f_n)$ is a convergent sequence of polynomials on D. prove that the limit function f is also a polynomial; or find a counterexample. Is this equivalent to "the limit of a sequence of ...
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Pointwise and uniform convergence application

Let $(f_{n})_{n}$ be a sequence of functions $f_{n}: \mathbb{R} \to \mathbb{R}$. They converge pointwise to a function $f: \mathbb{R} \to \mathbb{R}$. We know that the convergence is uniform on ...
83 views