Prove from the definition that $l^{2}(\mathbb{N})=\{(x_{1},x_{2},...)|x_{i} \in \mathbb{C}, \sum_{n=1}^{\infty}|x_{n}|<\infty\}$ is complete and separable.
Well, I need help solving this. So to get started, I want to know this: my prompt says to prove this from the definition (but, definition of what)? definition of...the space of square summable sequences...in the natural numbers? What would this mean, exactly?
$x\in \ell^2$ has the form $\{x_n\}_{n=1}^{\infty}$ where $\|x\|^2 = \sum_n |x_n|^2 < \infty$. For $\epsilon > 0$ there is an element of $\ell^2$ with only finitely many non-zero entries, and those entries have rational components only. To show this, first select $N$ such that $$ \sum_{n=N+1}|x_n|^2 < \frac{\epsilon^2}{4} $$ Then choose $\{ x_n '\}_{n=1}^{N}$ that have rational real and imaginary components such that $\sum_{n=1}^{N}|x_n-x_n'|^2 < \frac{\epsilon^2}{4}$. Then $$ \|\{ x_1,x_2,x_3,\cdots \} - \{ x_1',x_2',\cdots,x_N',0,0,0,\cdots\}\| < \epsilon. $$ Finally, show that the set of all such approximating elements is a countable set, thereby proving the existence of a countable dense subset for $\ell^2$.