I know that $\displaystyle l^\infty/c_0$ is complete.
Prove that $\displaystyle ||[a_n]||=\limsup|a_n|$

Where the norm $\displaystyle ||[a_n]||= \inf_{y_n \in c_0} ||a_n + y_n||_{\infty}$


I tried to find a relationship knowing that

$\displaystyle ||[a_n]||= \inf_{y_n \in c_0} ||a_n + y_n||_{\infty}= \inf_{y_n \in c_0}(\sup_{n \in N}|a_n+y_n|)$

and that

$\displaystyle \limsup|a_n| = \inf \sup |a_n|$

Any ideas?

Thanks.