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 $ ?