A question on $L^{p}$ spaces Consider the space of essentially bounded functions from $\mathbb{R}^{n}$ to $\mathbb{C}$ , denoted by  $L^{\infty}(\mathbb{R}^{n})$.
Note that the $p$-norm is defined by $\| g\|_{p}=[\int_{\mathbb{R}^{n}}|g|^{p}]^{1/p}$, for $p\geq1$.
Let $g\in L^{1}(\mathbb{R}^{n}) \cap L^{\infty}(\mathbb{R}^{n})$. Show that $\| g\|_{p}$ is finite for all $p\geq1$.
Any hint or help will be greatly appreciated.
 A: Actually with the same effort, we can prove a more general result

Let $r \geq 1$ and $g\in L^{r}(\mathbb{R}^{n}) \cap L^{\infty}(\mathbb{R}^{n})$. Then  $\| g\|_{p}$ is finite for all $p\geq r$.

Proof:
If $p = +\infty$ the result is trivial. So, suppose that  $1 \leqslant r <\infty$ and $1 \leqslant p <\infty$.
Since $g\in  L^{\infty}(\mathbb{R}^{n})$, there is $M >0$ such that $|g| < M$ a.e.. So,
for $p$ such that $r \leqslant p <\infty$, we have
$$ |g|^p < M^{p-r}|g|^r$$
So
$$ \int |g|^p < M^{p-r}\int |g|^r = M^{p-1}\|g\|_r^r $$
Since $g\in L^{r}(\mathbb{R}^{n})$, we have that $\|g\|_r<\infty$. So we have
$$\|g\|_p^p= \int |g|^p < M^{p-r}\int |g|^r = M^{p-r}\|g\|_r^r <\infty$$
So $\|g\|_p<\infty $.
A: I think the answer given by Ramiro completely solves your question. I would like to state a more general result as follows.

Let $\Omega\subset \mathbb{R}^n$,$1\leq p\leq r\leq q <\infty$. If $f\in L^p(\Omega)\cup L^q(\Omega)$, then $f\in L^r(\Omega)$.

Proof. We have the chain of equalities.
\begin{align}
&\int_\Omega|f|^r\mathrm{d}x\\
=&\int_{\Omega\cup \{|f|\geq 1\}}|f|^r \mathrm{d}x+ \int_{\Omega\cup \{|f|<1\}}|f|^r \mathrm{d}x\\
=& \int_{\Omega\cup \{|f|\geq 1\}}|f|^q \mathrm{d}x+ \int_{\Omega\cup \{|f|<1\}}|f|^p \mathrm{d}x\\
\leq&\|f\|_q^q+\|f\|_p^p<\infty
\end{align}
