Finding a functional on $l^{\infty}$ with certain properties

Question

My task is to show that there exist a linear functional $\varphi$ on $l^{\infty}$ such that:

1. $\varphi(x) = \lim x_n$ for $x \in C$ ($C$ denotes space of sequences with limits),
2. $\vert \vert\varphi \rvert\rvert = 1$.

Let's define $\widetilde{\varphi}$ on $C$ in the following way: $\widetilde{\varphi}(x) = \lim x_n$. It obvious that $$|\widetilde{\varphi}(x)| \le \sup |x_n| = ||x||$$ thus according to Hahn-Banach's theorem there exist a functional $\varphi$ on $l^{\infty}$ such that $|\varphi(x)| \le ||x||$. But how can I show that $||\varphi|| = 1$?

I would appreciate any tips.

Answer 1

How about inserting $x_n = 1$ for every $n$?

Answer 2

If $x=(1,1,\ldots)$ then $\|x\|=1$ and $\tilde \phi(x)=1$ so $\|\tilde \phi\|\ge 1$