Spaces $l(p_k)$ of finite series

Question

In the Elements of Functional Analysis by Maddox there is a problem who stays that given a $p=(p_k)$ sequence of positive reals we define $$l(p):=\{(x_k):\sum |x_k|^{p_k}<\infty\},$$ the problem asks for to prove that if $l(p)$ is a linear space then the sequence $p$ is bounded. I have tried this by using contrapositive and to prove this set is not closed by any of the linear operations.