$(f_n)$ converges uniformly $\iff \|f_n-f\|_\infty=\sup\{|f_n(x)-f(x)|;\ x\in I\}$
94066
Prove that: {$f_n$} converges unifromly to $f$ on $E$ as $n \rightarrow \infty$ if and only if $\lim_{n\rightarrow \infty} ||f_n-f||_u = 0$
133451
To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
60282
Prove that $f_n$ converges uniformly on $\mathbb{R}$
85268
Convergence of $\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$
35535
Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?
15579
Show that $f$ is uniformly continuous if limit exists
27153
Definition of convergence of $\sum_{i=-\infty}^\infty a_i$
131425
Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$
62892
Uniform convergence of $\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$
100580
Proof that a sequence converges.
134637
Exchanging of limits in case of uniform convergence.
66198
Sufficient condition for convergence of a real sequence
107399
A sequence and convergence
119348
Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise
98762
$f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$?
48500
Proof of convergence of sequences
150305
$f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent