Inequalities in sequences

Inequalities in sequences
Consider any scalar sequences $$\{x_k\}_{k=1}^\infty, \{y_k\}_{k=1}^\infty$$, the following inequalities hold (Th. 1.7.3., L1.7.4 )

(i) (Hölder's inequality) For any numberes $$p,q \in (1,\infty)$$ with $$1/p+1/q=1$$:

$$\sum_{k=1}^\infty|x_k y_k|\le \left(\sum_{k=1}^\infty |x_k|^p\right)^{1/p} \left(\sum_{k=1}^\infty |y_k|^q \right)^{1/q}$$

(ii) (Minkowski's inequality) For any $$p \in [1,\infty)$$

$$\left( \sum_{k=1}^{\infty} |x_k+y_k|^{p} \right)^{1/p} \le \left( \sum_{k=1}^\infty |x_k|^p\right)^{1/p}+\left( \sum_{k=1}^\infty |y_k|^p\right)^{1/p}$$

(iii) (Absolute convergence implies convergence)

$$\left| \sum_{k=1}^\infty x_k\right|\le \sum_{k=1}^\infty |x_k|$$