Given the sequence spaces $\ell^p$ that are defined as:
$$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$
for $\infty > p ≥ 1$, I'm trying to show that $\ell^p$ becomes a normed space with the norm:
$$||a||_p = \left(\sum_{n=0}^\infty |a_n|^p\right)^{1/p}$$
Now it's not to hard to see why $||a||_p$ is positive, definite and homogenous, but I've been struggling with the triangle inequality. How can this be shown? Could this be an application of the Cauchy-Schwarz-inequality or the Minowski-inequality? Or don't I need any of those to show that the triangle inequality is indeed valid? Thanks in advance!