Let $\displaystyle \ell^1(\mathbb{N},\mathbb{R})$ denote the space of sequences $\displaystyle x(j)$ such that $\displaystyle \sum_{j=1}^{\infty}|x(j)|<\infty$, equipped with the norm $\displaystyle \|x\|_1=\sum_{j=1}^{\infty}|x(j)|$. Let $\displaystyle \|x\|_2$ denote the usual norm $\displaystyle (\sum_{j=1}^{\infty}|x(j)|^2)^\frac{1}{2}$ on $\displaystyle \ell^2(\mathbb{N},\mathbb{R})$.

Show that $\displaystyle \|x\|_2\leq\|x\|_1$, and hence deduce that $\displaystyle \ell^1(\mathbb{N},\mathbb{R})\subseteq\ell^2(\math bb{N},\mathbb{R})$.

Any help would be appreciated. Thanks in advance