$\ell_{p}$ space closed to addition

Question

I'm trying to show that $\ell_{p}$ is a vector space for any $1\leqslant p<\infty$ . So given two infinite series $\left(x_{n}\right)_{n=1}^{\infty}$ and $\left(y_{n}\right)_{n=1}^{\infty}$ such such that ${\displaystyle \sum_{n=1}^{\infty}\left|x_{n}\right|^{p}}$ and ${\displaystyle \sum_{n=1}^{\infty}\left|y_{n}\right|^{p}}$ converge and $\alpha,\beta\in\mathbb{R}$ I want to show that $$\left[\alpha\left(x_{n}\right)_{n=1}^{\infty}+\beta\left(y_{n}\right)_{n=1}^{\infty}\right]\in\ell_{p}$$ To do that I need to show that $${\displaystyle \sum_{n=1}^{\infty}\left|\alpha x_{n}+\beta y_{n}\right|^{p}}<\infty$$ For some reason I'm finding this quite hard, I've been unsuccesful in bounding this sum from above by the two convergent series I started with.

Answer 1

You are actually trying to prove it's a subspace of a bigger vector space, for which we know it's already one. For example, the space of real sequences. (otherwise, we have to check all the axioms).

Hint: use and show the inequality $(a+b)^p\leqslant 2^{p-1}(a^p+b^p)$ for non-negative $a$ and $b$.

Answer 2

You could look at the proof of the Minkowski inequality (the sequence case).