# 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
0answers
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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 $x$ ...
3answers
75 views

### 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,...
0answers
24 views

### $F$ 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 $U\subset \Omega$. ...
1answer
93 views

### 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 ...
0answers
391 views

### 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 ...
2answers
118 views

### 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....
0answers
27 views

### Show that $f_n (x)$ converges to $f(x)=0 \ \forall x$

$f_n$ is defined on $[0,1] \ \ \forall n \in \mathbb{N}$, $f_n(x)=\begin{cases} n(1-nx) &\mbox{if } \ 0 \leq x < \frac{1}{n} \\ 0 & \mbox{if } \ \frac{1}{n} \leq x \leq 1 \end{cases}$ ...
0answers
40 views

### 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 ...
1answer
47 views

### 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 ...
2answers
64 views

### 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.
1answer
246 views

### 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 ...
2answers
945 views

1answer
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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 ...
4answers
83 views

1answer
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0answers
57 views