I'm studying functional analsyis and I keep wondering if there are some inclusions between the spac eof the continuous function and $L^p$ spaces (it will appear soon a similar question regarding $C_b$ and $C_c$, but let's focus on the supposed simplest case).
So the question is: consider $\Omega$ a bounded open set in $\mathbb{R}^n$, and consider respectively $C^0(\bar{\Omega})$ and $L^p(\Omega)$ (whhere $p\in [1,+\infty]$).
My idea: by the nesting property of $L^p(\Omega)$ ($\Omega$ by construction has finite Lebesgue measure), if $p<q$ then $L^q(\Omega)\subset L^p(\Omega)$. Therefore it is sufficient to prove it for $L^{\infty}(\Omega)$. But now it is quite simple to see that if $f\in C^0(\bar\Omega)$, then $\left\lVert f\right\rVert_{\infty,\bar{\Omega}}=\left\lVert f\right\rVert_{L^{\infty}}$. Thus we have the inclusion $C^0(\bar\Omega)\subset L^{\infty}(\Omega)$, so $C^0(\bar\Omega)\subset L^p(\Omega)$.
Is it true? Am i missing something very simple? I tried to look on literature about these result but maybe it's obvious or totally wrong, we'll see.
Finally, I was also wondering if by this I can easily conclude that $C^k(\bar\Omega)\subset L^p(\Omega)$, as $C^k(\bar\Omega)\subset C^0(\bar\Omega)$ by construction. If the above computations are correct it should immediately follow, but I ask for sure.
Any hint, correction, solution or reference would be much appreciate, thanks in advance.
This follows because a continuous function on a closed and bounded set is bounded. So, for any $f \in C(\overline{\Omega})$, we have $$ \biggl(\int_{\Omega} |f(x)|^p dx\biggr)^{1/p} \leq \mathcal{L}^n(\Omega)^{1/p} (\sup_{x \in \Omega}|f(x)|) < \infty. $$