from what I understand, it is easy to show that $\ell^p$ space is linear, without the famous inequalities (Minkowsky, Holder, etc.). The fact that is a metric space is not of interest for the moment. The question is first is about linearity.
What I particulary want to know is how to show that if $x,y\in \ell^p$, then also $x+y\in\ell^p$ for $p>1$. Thank you in advance.
If you are not interested in the triangle inequality, you may substitute Minkowski's inquality with the first step of its proof:
For $1 \le p < \infty$ the function $x \mapsto |x|^p$ is convex on $\mathbb{R}$. Hence, for $x,y \in \mathbb{R}$, we have $$|x + y|^p = \Big| \frac12 \, 2\,x + \frac12 \, 2 \, y|^p \le \frac12 \,|2\,x|^p + \frac12 \,|2\,y|^p=2^{p-1}\,(|x|^p + |y|^p).$$ Hence, for $x,y \in \ell^p$ we find $$\| x + y\|_p^p = \sum_{i=1}^\infty |x_i + y_i|^p \le 2^{p-1} \, \sum_{i=1}^\infty |x_i|^p + |y_i|^p = 2^{p-1} \, (\|x\|_p^p + \|y\|_p^p).$$
The case $p = \infty$ is trivial.