Doesn't this $L^p$ norm estimate for all $p$ give me an $L^\infty$ bound?

Question

Let $r_n \to \infty$ as $n \to \infty$. We have that $$\lVert v \rVert_{L^{r_n}(\Omega)} \leq C\lVert v \rVert_{L^{r_0}(\Omega)} < \infty$$ for all $n$, where $C$ is independent of $v$ and $n$.

Can I not just take $n \to \infty$ in this inequality to conclude that $v \in L^\infty(\Omega)$? The reason I ask is, that the author of this paper (see page 300, where my $v$ is his $u^+$) proves after proving the above quoted statement that $v \in L^\infty(\Omega)$ by a proof by contradiction.

Answer 1

You totally can, since $C$ is independent of $n$. Sometimes not doing so is illustratively useful, but you can.