Minkowski's inequality says the following: For every sequence of scalars $a = (a_i)$ and $b = (b_i)$, and for $1 \leq p \leq \infty$ we have: $||a+b||_{p} \leq ||a||_{p}+ ||b||_{p}$. Note that $||x||_{p} = \left(\smash{\sum\limits_{i=1}^{\infty}} |x_i|^{p}\right)^{1/p}$. This is how I tried proving it: \begin{align*} ||a+b||^{p} &= \sum |a_k+b_k|^{p}\\\ &\leq \sum(|a_k|+|b_k|)^{p}\\\ &= \sum(|a_k|+|b_k|)^{p-1}|a_k|+ \sum(|a_k|+|b_k|)^{p-1}|b_k|. \end{align*}
From here, how would you proceed? I know that you need to use Hölder's inequality. So maybe we can bound both the sums on the RHS since they are products.