$ p \geq q \geq 1$, then $L^p_{[a,b]} \subset L^q_{[a,b]}$

I want to prove it with Holder inequality for integrals. However I am not sure how to proceed. This is what I did:

Let $f \in L^p_{[a,b]}$. Then I can write: $$ \int_a^b \vert f(x) \vert^p dx < \infty $$

$$ \int_a^b \vert f(x) \vert^p dx = \int_a^b \vert f(x) \vert^{p-q+q} dx $$ $$ \int_a^b \vert f(x) \vert^{p+q} \vert f(x) \vert^{-q}dx $$ But from here I do not know how to proceed. I am not even sure this is the right way to approach the problem.

EDIT: As suggested in the comments I tried to prove that $\frac{1}{p} = \frac{1}{q} + \frac{1}{r}$ and Holder inequality gives: $\Vert fg \Vert _p \leq \Vert f \Vert_q \Vert g \Vert_r$

By holder inequality and the equality that I have written before I can write: $$ \Vert fg \Vert_{\frac{1}{p}} \leq \Vert f \Vert_{\frac{1}{q}} \Vert g \Vert_{\frac{1}{p}} $$

Then the inequality follows. However I am struggling to see how I can write $\Vert f \Vert_q$ less than something that converges.