I have the normed space $ \ell^1 =\{(x\_n) : \sum\_{n=1}^\infty |x\_n| < \infty\}$ with the norm defined by $\|x\|\_1 = \sum\_{n=1}^\infty|x\_n| $, and a function
$$f(x):=\sum\_{n=1}^\infty x\_n\sin(n) .$$ 
I wanna prove that it is continuous from $( \ell^1, \|\cdot\|)$ to $\mathbb{R}$ with usual metric.

Here's what I've done: let $x,y \in \ell^1$
$$\begin{align}
|f(y) - f(x)|&= |\sum\_{n=1}^\infty y\_n\sin(n) - \sum\_{n=1}^\infty x\_n\sin(n) | \\
&= |\sum\_{n=1}^\infty (y\_n -x\_n) \sin(n) | \\
&\leq | \sum\_{n=1}^\infty (y\_n -x\_n)|\\
&= |\sum\_{n=1}^\infty y\_n - \sum\_{n=1}^\infty x\_n |\\
& = |\|y\| - \|x\|| \\
& \leq\|x-y\|.
\end{align}$$
So I got $ d\_R \leq d\_{\ell^1} $.

I think the main work is done. I just don't know how to fit it together? Using the definition of continuous functions, should I choose, $\epsilon \leq \delta $ ?