Continuity of linear maps $l^\infty\rightarrow l^\infty$

Question

Let $C=(c_1,c_2,...)\in l^1$ and $A:l^\infty\rightarrow l^\infty$ a linear map with $A(x)=(c_1x_1,c_1x_1+c_2x_2,c_1x_1+c_2x_2+c_3x_3,...)$ for $x=(x_1,x_2,...)\in l^\infty$.

1. How do I prove that $A$ is continuous?
2. Let $c=(c_1,c_2,...)$ be a sequence in a field $k$ such that $(c_1x_1,c_1x_1+c_2x_2,c_1x_1+c_2x_2+c_3x_3,...)\in l^\infty$ for $x\in l^\infty$.
   How do I prove that $c\in l^1$?

What I know:

I know that $l^p$ is the vector space of bounded sequences $\{x_i\}$ in $k$ such that $\sum |x_i|^p<\infty$. On $l^1$ the norm is $||x||_1=\sum|x_i|$ and on $l^\infty$ the norm is $||x||=\sup\{|x_i|\}$.
Furthermore I know that one of the definitions of continuity (and probably the one I should use here) is that there exists a constant $\alpha$ such that $||A(x)||\leq \alpha||x||$ for all $x\in l^\infty$. Thus what I'm looking to prove is that $\sup\{|A(x)_i|\}\leq \alpha\sup\{|x_i|\}$. How do I do this?

Answer 1

$\|A(x)\|_\infty=\sup|\sum_{i=1}^n c_i x_i|\leq\sup\sum_{i=1}^n|c_i||x_i|\leq\sup\sum_{i=1}^n|c_i|\cdot\|x\|_\infty=\|C\|_1\cdot\|x\|_\infty$.

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Edit. Answer to Question 2.
I assume the field $k$ is restricted to $\mathbb{R}$ and $\mathbb{C}$, otherwise it's not easy to define norms. In both cases, for any number $t$ in the field $k$ there exists a number $f(t)$ such that $|f(t)|=1,tf(t)=|t|$. More precisely, one may choose $f(t)=\operatorname{sgn}(t)$ if $k=\mathbb{R}$, and $f(t)=e^{-i\arg t}$ if $k=\mathbb{C}$.
Now take $x=(f(c_1),f(c_2),\cdots)$. It's easy to see $\|x\|_\infty=1$, hence $x\in l^\infty$. But due to the condition we have $\sup|\sum_{i=1}^n c_i x_i|=\sup\sum_{i=1}^n|c_i|=\sum_{i=1}^\infty|c_i|<\infty$, which shows $c\in l^1$.