Show that the linear functional is bounded and find its norm.

Question

Let us define a functional on a space of convergent sequences $c$ with a norm $\left\Vert x \right\Vert_\infty =\sup_{1\leq k\leq\infty}\left|x_{k}\right|$ by an equation:
$$f(x)=\lim_{n\rightarrow\infty}x_{n}-\sum_{n=1}^{\infty}2^{-n}x_{n}$$
where $x=(x_{k})_{k=1}^{\infty}\in c$. Show that the functional given above is bounded.
An attempt to solve
Since there is a minus sign between the limit and the sum, triangle inequality leads not in the direction I want:
$$\left|f(x)\right|=\left|\lim_{n\rightarrow\infty}x_{n}-\sum_{n=1}^{\infty}2^{-n}x_{n}\right|\geq\left|\lim_{n\rightarrow\infty}x_{n}\right|-\left|\sum_{n=1}^{\infty}2^{-n}x_{n}\right|$$
On the other hand each of the elements can be clearly analyzed. We have $\left|\lim_{n\rightarrow\infty}x_{n}\right|\leq\sup_{1\leq k\leq\infty}\left|x_{n}\right|$ and
$$\sum_{n=1}^{\infty}2^{-n}x_{n}\leq\sup_{1\leq k\leq\infty}\left|x_{k}\right|\cdot\sum_{n=1}^{\infty}2^{-n}=\sup_{1\leq k\leq\infty}\left|x_{k}\right|$$
(At least I think it makes sense). We could therefore go on and write:
$$\left|f(x)\right|\leq\sup_{1\leq k\leq\infty}\left|x_{k}\right|+\sup_{1\leq k\leq\infty}\left|x_{k}\right|=2\left\Vert x\right\Vert $$
which means $\left|f(x)\right|\leq C\left\Vert x\right\Vert $ where $C=2$ for all $x\in c$ and thus prove functional is bounded. To find the norm I go for the definition $\left\Vert f\right\Vert =\sup_{\left\Vert x\right\Vert =1}\left|f(x)\right|$ but it leaves me wondering how to calculate it. Does my attempt make any sense?