Let $Z$ be an integrable random variable on filtered probability space $(\Omega , \mathscr F, (\mathscr {F_n})_{\{n \in \mathbb{N}\}}, \mathbb P)$
Define $Z_{n} := E[Z|\mathscr {F_n}]$. Show that $Z_n$ is integrable i.e. $E[|Z_n|] < \infty$
What I tried:
$E[|Z_n|] = E[|E[Z|\mathscr {F_n}]|] \le E[E[|Z||\mathscr {F_n}]] = E[|Z|] < \infty$ QED
Did I do that inequality* right? Any mistakes?
*It's the Triangle Inequality for Integrals.