real analysis - Prove that $lim_{pto infty} |u|_p = |u|_{infty}$

Let $(X,\mathcal{A}, \mu)$ be any measure space and let $u \in \bigcap_{p\in [1,\infty]} \mathcal{L}^p(\mu)$. Then

$$\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}.$$

I have already proved the inequality $\|u\|_{\infty} \leq \liminf_{p\to \infty} \| u \|_p$, so I still need to prove $\limsup_{p\to \infty} \| u \|_p \leq \| u \|_{\infty}$.

Following a hint, I have proved that for any sequence $q_n \to \infty$ one has

$$\|u\|_{p+q_n} \leq \|u\|_{\infty}^{q_n/(p+q_n)} \cdot \|u\|_p^{p/(p+q_n)}.$$

How can we conclude from this that $\limsup_{p\to \infty} \| u \|_p \leq \| u \|_{\infty}$?