Dual space of $l^p$ is $l^q$.

Question

I was reading functional analysis from Kreyszig. While proving that dual space of $l^p$ is $l^q$, I came across a doubt. I have attached the screenshot. In this they are applying f to $x_n$ but for that $x_n$ ,must be in $l^p$ because f is an element of dual space of $l^p$. I tried proving $x_n$ is in $l^p$ so I take $|\zeta_1|^p+...+|\zeta_n|^p$which is simplified to $|\gamma_1|^q+...+|\gamma_n|^q$ but now how to proceed?