How is it proven that $\sum_{n = 1}^{\infty} |a_{n} + b_{n}|^p < \infty$ under the conditions below?

Question

Let us assume that $1 \leq p < \infty$. Also, we assume that $\forall n \in \mathbb{N}$, $a_{n}, b_{n} \in \mathbb{R}$ and $\sum_{n = 1}^{\infty} |a_{n}|^p < \infty, \sum_{n = 1}^{\infty} |b_{n}|^p < \infty$. Considering all the conditions I described above, how is it shown that $\sum_{n = 1}^{\infty} |a_{n} + b_{n}|^p < \infty$? This lemma seems crucial for proving the Minkowski's inequality.

Answer 1

The trick is to use convexity of $|t|^p$. We have $$ \sum_{n=1}^\infty |a_n + b_n|^p = 2^p\sum_{n=1}^\infty \left|\frac{a_n + b_n}2\right|^p \le 2^{p-1} \sum_{n=1}^\infty |a_n|^p + 2^{p-1} \sum_{n=1}^\infty |b_n|^p < \infty. $$