Prove that $(\ell^p, \|\cdot\|_p)$ is complete.
My attempt $:$ Let $\sum\limits_{n=0}^{\infty} x^{(n)}$ be an absolutely summable series in $\ell^p.$ Let $x^{(n)} = \left (x_j^{(n)} \right )_{j \geq 0}.$ Need to show that $\sum\limits_{n=0}^{\infty} x^{(n)}$ converges in $\ell^p.$ Let $M = \sum\limits_{n=0}^{\infty} \left \|x^{(n)} \right \|_p.$ Let $(y^{(n)})_{n \geq 1}$ be the sequence of partial sums of the series $\sum\limits_{n=0}^{\infty} x^{(n)}.$ So for each $n \geq 0$ we have $$y^{(n)} = \sum\limits_{k=0}^{n} x^{(k)} = \left (\sum\limits_{k=0}^{n} x_j^{(k)} \right )_{j \geq 0} = \left (y_j^{(n)} \right )_{j \geq 0}.$$
where $y_j^{(n)} = \sum\limits_{k=0}^{n} x_j^{(k)}, j \geq 0.$ Now for each $j \geq 0$ consider the sequence $\left (z_j^{(n)} \right )_{n \geq 0}$ defined by $$z_j^{(n)} = \sum\limits_{k=0}^{n} \left |x_j^{(k)} \right |,\ n \geq 0.$$ Then it is easy to see that $\left (z_j^{(n)} \right )_{n \geq 0}$ is increasing and bounded above by $M$ for all $j \geq 0$ and hence it is convergent for all $j \geq 0.$ Let $z_j : = \lim\limits_{n \to \infty} z_j^{(n)},\ j \geq 0.$ For each $n \geq 0$ define the sequence $\left |x^{(n)} \right | : = \left (\left |x_j^{(n)} \right | \right )_{j \geq 0}.$ Let $z^{(n)} : = \sum\limits_{k=0}^{n} \left |x^{(k)} \right | = \left (\sum\limits_{k=0}^{n} \left |x_j^{(k)} \right | \right )_{j \geq 0} = \left (z_j^{(n)} \right )_{j \geq 0},\ n \geq 0.$ Then by Minkowski's inequality it follows that $$\left \|z^{(n)} \right \|_p \leq \sum\limits_{k=0}^{n} \left \|x^{(k)} \right \|_p \leq M.$$ Now for each $j \geq 0$ we have $\left (z_j^{(n)} \right )_{n \geq 0}$ increases to $z_j.$ So by MCT it follows that $$\sum\limits_{j=0}^{\infty} z_j^p = \lim\limits_{n \to \infty} \sum\limits_{j=0}^{\infty} {z_j^{(n)}}^p = \lim\limits_{n \to \infty} \left \|z^{(n)} \right \|_p^{p} \leq M^p.$$ This shows that $\|z\|_p \leq M,$ where $z = (z_j)_{j \geq 0}.$ Hence $z \in \ell^p.$ Now since $\left (z_j^{(n)} \right )_{n \geq 0}$ is convergent for each $j \geq 0,$ it follows that $\left (y_j^{(n)} \right )_{n \geq 0}$ is convergent for each $j \geq 0.$ Let $y_j = \lim\limits_{n \to \infty} y_j^{(n)},\ j \geq 0.$ Let $y = (y_j)_{j \geq 0}.$ If we can prove that $y \in \ell^p$ and $\left \|y^{(n)} - y \right \|_p \to 0$ as $n \to \infty$ then we are through. Now $$\left |y_j^{(n)} \right | \leq \sum\limits_{k=0}^{n} \left |x_j^{(k)} \right | = z_j^{(n)}.$$ Hence we have $$\left |y_j^{(n)} \right |^p \leq {z_j^{(n)}}^p.$$ Since $\left \|z^{(n)} \right \|_p \lt \infty$ it follows from DCT that $y \in \ell^p$ and $$\left \|y^{(n)} - y \right \|_p \to 0\ \text {as}\ n \to \infty.$$
Is my attempt correct at all? Please check it.
Thanks in advance.