Dual space of $\mathbb{R}^{\infty}$ with $l^2$ norm equals $l^2$

Question

As usual, denote $\mathbb{R}^{\infty}=\{ x : \mathbb{N} \rightarrow \mathbb{R} : \exists N \ \ s.t. \ x_n=0 \ \ \forall n \ge N \}$; $l^2= \{ x : \mathbb{N} \rightarrow \mathbb{R} : \sum_{n=1}^{+\infty} |x_n|^2 < +\infty \}$ and $\| x \|_{l^2}= \sqrt{\sum_{n=1}^{+\infty} |x_n|^2 }$. I must prove that $(\mathbb{R}^{\infty},\| \ \|_{l^2})' \equiv l^2$. A hint for this exercise says: prove that for each $f \in (\mathbb{R}^{\infty},\| \ \|_{l^2})'$ there exists $y \in l^2$ such that $f(x)=\sum_{n=1}^{+\infty} y_n x_n $ for every $x \in l^2$. I basically have two issues: first of all I cannot understand why it sufficies to prove the previous assertion to conclude; when it comes to the assertion itself, the only two ideas I have is to use the fact that $(\mathbb{R}^{\infty},\| \ \|_{l^2}) \subset (l^2, \| \ \|_{l^2})$, which is an Hilbert space, then apply Riesz's representation to an extension of $f$ and finally restrict again to $(\mathbb{R}^{\infty},\| \ \|_{l^2})$ or try to use the density of $\mathbb{R}^{\infty}$ in $l^2$ (but I don't know how). Do these ideas make sense? Any hint or comment is very appreciated.