I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\mathbb{N})$. My attempt is the following:
By definition, the quotient norm $\| \cdot \|_{\ell^{\infty} / c_0}$ is defined as $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = d( x + c_0 (\mathbb{N}), 0 + c_0 (\mathbb{N}) ), $$ which can be re-written as $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = \inf \{ \| x + y \|_{\infty} \; | \; y \in c_0 (\mathbb{N}) \}. $$ Since $0 = (0,0,...) \in c_0 (\mathbb{N})$, it follows that $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} \leq \| x \|_{\infty} = \sup_{n \in \mathbb{N}} |x_n|. $$
I thought about using Bolzano-Weiserstrass' Theorem at a certain point, but I am not sure how to obtain the wished equality from here.
Any help is appreciated. Thanks in advance.
I'll prove it showing both inequality's, first since $y_n = (x_1,x_2,...,x_n,0,0,...)\in c_0$ for all $n\in\mathbb{N}$ one has $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} \leq \inf_n \| x-y_n\|_\infty = \inf_n \sup_{m\geq n} |x_n| = \limsup_n |x_n|. $$ On the other hand by definition, for all $y=(y_1,y_2,y_3,...)\in c_0$ there is an $n\geq 1$ such that $y_m =0$ for all $m\geq n$, hence $$ \| x+y\|_\infty \geq \sup_{m\geq n}|x_n| \geq \limsup_n |x_n| $$
$$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = \inf \{\|x+y\|_\infty\;|\; y\in c_0\} \geq \limsup_n |x_n|. $$