Suppose $p(n)>0$, prove that $X$ is a linear space if and only if $\sup_n p(n)<\infty$.

Question

$X=\{ (x_k)_{k=1}^{\infty}:\sum_1^{\infty}\mid x_k \mid^{p(n)} \}$

I am able to show that, if $\sup_n p(n)=N<\infty$, then for a scalar $\alpha$, we have $\sum_1^{\infty}\mid \alpha x_k \mid^{p(n)}\leq \mid \alpha \mid^N\sum_1^{\infty}\mid x_k \mid^{p(n)} $. However i cannot show that $\sum_1^{\infty}(x_k+y_k)^{p(n)}<\infty$. Any hints or tips would be much appreciated !