Monotonicity of $\mathcal{l^p}$ spaces using only Hoelder inequality

Question

For $p > 0$, let $\ell^p$ be the space of sequences for which  $$\sum_{i=1}^{\infty} |a_i|^p$$  is finite ($a_i \in \mathbb{R}$). It is well-known that, for $q > p$, $$\ell^p \subset \ell^q.$$ Can this be shown only using the Hoelder inequality for such sequence spaces (which I know/can prove, so it can be assumed)? I remember the Jensen Inequality proof for the similar statement for the monotonicity for $\mathcal{L_p}$ spaces, but there should be a clever way of just using Hoelder - I think -, and I am blanking on how to do it. 

Answer 1

You do not need Holder. Take $a \in \ell^p$. Then since the sum $\sum |a_i|^p$ converges, we have $a_i \to 0$, meaning for large enough $i$, $|a_i| < 1$. This means $|a_i|^q < |a_i|^p$ for large enough $i$, so $\sum |a_i|^q$ also converges.