We denote by $\mathbb{K}$ either real or complex space. And we consider $c_{0} = \{(x_{j})_{j \in \mathbb{N}} |(\forall j \in \mathbb{N})(x_{j} \in \mathbb{K}) \land (\lim_{j\to\infty}x_{j} = 0) \}$, the scalar null sequences. I want to show that $\ell^{2} = \{(x_{j})_{j \in \mathbb{N}} | (\forall j \in \mathbb{N})(x_{j} \in \mathbb{K}) \land (\sum_{j=1}^{ \infty} |x_{j}|^{2} < \infty) \}$ is contained in $c_{0}$. My approach is this:
This is new to me, but I read that $\sum_{j=1}^{ \infty} |x_{j}|^{2} < \infty$ is another way of saying that this series converges.. So assuming that's true, given an arbitrary element of $\ell^{2}$, if $\sum_{j=1}^{ \infty} |x_{j}|^{2} < \infty$, then $\lim_{j\to\infty}x_{j}^{2}=0,$ and so $\lim_{j\to\infty}x_{j}=0.$ So the element belongs to $c_{0}$.