Prove that $c_0$ is complete by showing that $c_0$ is closed in $l_\infty$.
42108
Prove that $c_0$ is closed in $\ell^\infty$
30770
Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
82900
Show that Minkowski sum of a closed set and a compact set in $\mathbb{R}^n$ is closed.
137446
Show that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$.
111205
Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$
105046
Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$
154796
Completeness of sequence of reals with finitely many nonzero terms
57868
Prove $\tau$ is cofinite on $X$ iff $(X,\tau)$ is $T_1$-space and any infinite subset of $X$ is dense in $(X, \tau)$
73338
Prove that the closure of complement, is the complement of the interior
16826
prove that every infinite subset of compact set H has a limit point (Explanation)
156727
A step in the proof of Characterization of $W^{1,\infty}$
125962
The set $A'$ is closed in a $T_1$ space
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
74764
Dimension for a closed subspace of $C[0,1]$.
136851
Proof that the set of subsequential limits is closed
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
139595
Show that $(l_1)^* \cong l_{\infty}$