Question: I have a few questions regarding conjugates and moduli

1.) If $z=x+bi$, how would I simplify $\overline{z+a}$ or $\overline{z+i}$
Also, how would I simplify something like $\overline{2\overline{z}+5}$.

2.) Would $\overline{2} = 2$? In addition, would $\overline{3i}=-3i$?

3.) For the modulus, I have no idea how to simplify
$|3z-i|$, or $|z+2|$.

Thank you so much!

Answer: 
Assume that $X\_n, X$ are non-negative integrable RVs such that

$$ X\_n \to X \quad \text{a.s.}, \tag{1} $$
$$ \mathbb{E}[X\_n] \to \mathbb{E}[X] \tag{2}$$

hold. We also note that $\left|\max\{X, X\_n\} - X\right| \leq X\_n$. Then by the Fatou's Lemma,

\begin{align\*}
\mathbb{E}[X]
&= \mathbb{E}\left[\liminf\_{n\to\infty} (X\_n - \left|\max\{X, X\_n\} - X\right|)\right] \tag{by (1)} \\
&\leq \liminf\_{n\to\infty} \mathbb{E}\left[ X\_n - \left|\max\{X, X\_n\} - X\right| \right] \tag{by Fatou} \\
&= \mathbb{E}[X] - \limsup\_{n\to\infty} \mathbb{E}\left[ \left|\max\{X, X\_n\} - X\right| \right] \tag{by (2)} \\
\end{align\*}

This shows that $\limsup\_{n\to\infty} \mathbb{E}\left[ \left|\max\{X, X\_n\} - X\right| \right] = 0$, which in turn implies the desired $L^1$-convergence.
Note: $X\_n \geq 0$ was added later. 

This is false. Let $X\_n=nI\_{(0,\frac 1 n)} -nI\_{(-\frac 1 n, 0)}$ and $X=0$. Then $X\_n \to X$ almost surely, $EX\_n=0$ for all $n$, so $EX\_n \to EX$ but $\max \{X\_n,X\}=nI\_{(0,\frac 1 n)}$ which does not tend to $0$ in $L^{1}$.