supremum and summation Inequality

Question

I am trying to prove an Inequality $$ \sup_{k \ge 1} \mid x_k \, \mid ^{q-p} \le \, (\mid \sum_{k \ge 1} \mid x_k \mid ^p) ^{q-p \over p} $$ where $1 \le p \le q < \infty$ How should I proceed

Answer 1

It is a matter of the fact that \begin{align*} \sup|x_{k}|\leq\left(\sum|x_{k}|^{r}\right)^{1/r} \end{align*} for $r\geq 1$.

One shall note that \begin{align*} |x_{k}|=(|x_{k}|^{r})^{1/r}\leq\left(\sum|x_{k}|^{r}\right)^{1/r} \end{align*} for all $k$.