Is $\frac{nx}{n^{2}x^{2}+1}$ Cauchy in uniform norm (I guess it is not Cauchy but am not sure about my proof)?
18153
To prove that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not uniformly converge to $f(x) = 0$ on $[0,1]$
147841
Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$
9159
$L^p$-norm of a non-negative measurable function
143545
If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.
32742
The limit of $\frac{|A_n|}{n^2}$
33673
Proving $N(x) = (A(x,x))^\frac12$ is a norm in $V$ given that A is a positive definite bilinear form
26291
Does $\frac{nx}{1+n \sin(x)}$ converge uniformly on $[a,\pi/2]$ for all $a \in (0,\pi/2]$?
67485
The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets
85217
Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$
93806
$\|f\|_{L^{p}}=\sup\left\{\int_{{\bf R}^{n}}|f(x)g(x)|~dx:\|g\|_{L^{q}}\le1\right\}$
140379
Cauchy sequence in a discrete space
38185
Prove that $\{\cos(nx) : n=1, ... ,\infty \}$ is bounded but not totally bounded in $C([0,\pi])$
118903
The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product
108696
If $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence, then $(a_n ^2)_{n\in\mathbb{N}}$ is also a Cauchy sequence
149055
Uniform convergence of $\sum\limits_{k=1}^{\infty}\dfrac{kf(x^2)}{1+k^3f(x^2)^2}$
103124
Proving that $f(z)=\sum\limits_{n=1}^{\infty}\frac{z}{n(1+n|z|^{2})}$ converges uniformly on $\mathbb{C}$
157455
Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$