Checking the solution of the problem about metric function I would like to receive some help with the next problem:
I'm trying to proove the following:

Let's mark with $l^p$ ($1 \le p < \infty$) a set of all sequences $a = \{a_n\}$ of real numbers, such that series $\sum_{i = 1}^{\infty} |a_i|^p$ converges and define mapping $d: l^p \times l^p \mapsto [0, \infty)$ with
$$d(x, y) = \sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - y_i|^p}, \quad x, y \in l^p.$$
Proove that $d$ is metric on $l^p$.

I have a problem with the third step of my proof. I did this:
$$d(x, y) = \sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - y_i|^p} = \sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - z_i + z_i - y_i|^p} \le$$
$$\le \sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - z_i|^p} + \sqrt[p]{\sum\limits_{i = 1}^{\infty} |z_i - y_i|^p} = d(x, z) + d(z, y).$$
I'm trying to use Minkowski's inequality, but i'm not sure if i can use it if $i$ goes to $\infty$ instead to $n \in \mathbb{N}$.
Please, could you tell me if this step is correct and if it isn't, could you give me some advice about how to prove that, in this case, it's $d(x, y) \le d(x, z) + d(z, y)$.
 A: Yes, as you noted above, you can't apply Minkowski's inequality directly. But you can proceed as below :
Note that for any $n\in \Bbb N $,$$\sqrt[p]{\sum\limits_{i = 1}^{n} |x_i - y_i|^p}  \le
\sqrt[p]{\sum\limits_{i = 1}^{n} |x_i - z_i|^p} + \sqrt[p]{\sum\limits_{i = 1}^{n} |z_i - y_i|^p}\leq \sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - z_i|^p} + \sqrt[p]{\sum\limits_{i = 1}^{\infty} |z_i - y_i|^p}. $$
Since this is true for any  $n\in \Bbb N $, taking the limits will give you the required inequality. 

Let $S_n=\sqrt[p]{\sum\limits_{i = 1}^{n} |x_i - y_i|^p}$. Note that the sequence $\{S_n\}$ is an increasing sequence of non-negative real numbers and is bounded above by $$\sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - z_i|^p} + \sqrt[p]{\sum\limits_{i = 1}^{\infty} |z_i - y_i|^p}.$$ So $\{S_n\}$ converges to $$\lim_{n\to\infty}S_n=\sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - y_i|^p}$$ and the desired inequality holds (if you are still struggling, see what happens if  $$\sqrt[p]{\sum\limits_{i = 1}^{\infty} |x_i - z_i|^p} + \sqrt[p]{\sum\limits_{i = 1}^{\infty} |z_i - y_i|^p}\lt\lim_{n\to\infty}S_n).$$
