The dual space of the set of convergent sequences I'm trying to obtain the dual space of the set of convergent sequences. In proving this, I have to prove some propositions.
Let $c$ be the set of convergent complex sequences with $\|\cdot\|_\infty$ norm. First, I want to show that for any $\varphi\in c^*$, $\sum_{n=1}^\infty\varphi(e_n)$ is absolutely convergent, where $e_n$'s are canonical sequences in $\ell^\infty$.
Second, for each $\varphi\in c^*$, define $g_\varphi$ by
$g_\varphi(1)=\varphi(\mathbf{1})-\sum_{n=1}^\infty\varphi(e_n)$ and $g_\varphi(n)=\varphi(e_{n-1})$ for $n=2,3,\ldots$, where $\mathbf{1}=(1,1,\ldots)$. Then I want to show that $\|g_\varphi\|_1\leq \|\varphi\|$.
In second question, I showed that $g_\varphi$ is in $\ell^1$, but I failed to prove the inequality.

For the first part: write down 
$$
\sum_{n=1}^N |\varphi(e_n)| = \varphi\left(\sum_{n=1}^N c_n e_n\right) \tag{1}
$$
where $c_n$ is a unimodular complex number such that $c_n \varphi(e_n)=|\varphi(e_n)|$. Since the right hand side of $(1)$ is bounded by $\|\varphi\|$, the claim follows.
For the second part: consider the sequence 
$$x=(c_1,c_2,\dots,c_N, b,b,b,b,\dots)$$
where $b$ is a unimodular complex number such that $bg_\varphi(1)=|g_\varphi(1)|$. The inequality $|\varphi(x)|\le \|\varphi\|$ reads as
$$
\sum_{n=1}^N |\varphi(e_n)| +b\varphi\left(\sum_{n>N}e_n\right) \le \|\varphi\|
$$
Pass to the limit $N\to\infty$ to get the desired result.
