Show that norm in $ \ell_{\infty} /c_0 $ is $||\overline{x} ||= \text{lim sup}|x_i|$ for each $x=(x_i)_i \in \ell_{\infty} $.
I'm doing $$||x_n+c_0||= \text{inf}_{y_n \in c_0}||(x_n)+(y_n)||_{\infty}= \text{inf}_{y_n \in c_0 } \text{sup}_{n \in \mathbb{N}}|(x_n)+(y_n)|$$
I do not know what to do now. I thought to fix $y_n$ and use $y_n \to 0$, but I can not get anywhere.