Proving standard basis of $l^p$, $1<p<\infty$, to be shrinking

Question

The definition of shrinking basis is given inside the Wikipedia article, under the section "Schauder basis and Duality". It says that "if $1<p<\infty$, then the standard basis $(e_n)_n$ is shrinking". I need help to prove this statement.

Answer 1

You have to fix $y\in\ell^q$ and then estimate \begin{align} \Big|\sum_{m=n}^\infty x_my_m\Big| \end{align} for $x\in\ell^p$ with $\|x\|_p=1$. So \begin{align} \Big|\sum_{m=n}^\infty x_my_m\Big| \leq\Big(\sum_{m=n}^\infty |x_m|^p\Big)^{1/p} \Big(\sum_{m=n}^\infty |y_m|^q\Big)^{1/q} \leq\Big(\sum_{m=n}^\infty |y_m|^q\Big)^{1/q}. \end{align}