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

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


I tried to find a relationship knowing that

||[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

\limsup|a_n| = \inf \sup |a_n|

Any ideas?

Thanks.