$a \in \ell_q$ and $x \in \ell_p$. We also have $1/p + 1/q = 1$.
I want to show that $$ \cfrac{|a_ix_i|}{||x||_p ||a||_q} \leq \cfrac{1}{p}\left(\cfrac{|x_i|}{||x||_p} \right) +\cfrac{1}{q}\left(\cfrac{|a_i|}{||a||_q} \right) \\ \implies \cfrac{\sum\limits_{i=1}^{\infty}|a_ix_i|}{||x||_p||a||_q} \leq 1 $$
I started with
$$ \cfrac{\sum |a_ix_i|}{||x||_p ||a||_q} \leq \cfrac{1}{p}\left(\cfrac{\sum |x_i|}{||x||_p} \right) +\cfrac{1}{q}\left(\cfrac{\sum |a_i|}{||a||_q} \right) $$
and then I expanded the summations and wrote out the definition for the $p$-norm and $q$-norm but I don't see how to make the terms disappear. I know that the norms must converge to be a valid element of the space but this fact didn't help me reduce the RHS to 1. I know that somehow the right hand side should add to one but I'm stuggling to see it.
The comments make me think that there's a mistake in the homework. Here's an image of the problem.
Hint: Use Young's inequality for each summand.
This will look similar to your assumed inequality (maybe you meant to add the exponents so that it becomes Young's inequality).
edit: To address the edit of the question: Yes, there does seem to be a mistake. In the second part the exponents $p$ and $q$ are missing over the parts in the parentheses.