Show $f_n(x_n) \to f(x)$ if $f_n \to f$ uniformly and $x_n \to x$.
41482
$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
18153
To prove that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not uniformly converge to $f(x) = 0$ on $[0,1]$
60282
Prove that $f_n$ converges uniformly on $\mathbb{R}$
56567
Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$
45885
If there exist sequence such that $g(x_n)=f(x_{n+1})$, then we have $g(x_0)=f(x_0)$ for some $x_0$
113740
Find a uniformly continuous function such that $a_{n+1}=f(a_n)$
147661
Show that {$x_n$} is a convergent sequence if $d(x_{n+2}, x_{n+1})≤ rd(x_{n+1}, x_n)$
94737
How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$
93964
Proving that $f$ is a bijection from $N$x$N$ to $N$.
15727
show that if $f$ is uniformly differentiable then prove that $f'$ is continuous
40297
Proof of uniform limit of Continuous Functions
76485
$f_n(x) \to f(x)$ and $\int |f_n|^2 \, d\mu \to \int |f|^2 \, d\mu$. Use Egorov's theorem to show $f_n \to f$ in $L^2(d\mu)$.
124382
Showing that $f_n:[0,1]\to\mathbb{R}$ when $f_n=nx(1-x)^n$ converges pointwise but not uniformly
133451
To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
83036
f(g(x)) has a degree divisible by n
29287
Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
136842
$|x-f(x)|<5777$, Show that $f$ is surjective