I'm reading a particular proof of the $L^2$-bounded martingale convergence theorem, and there are three details that I don't follow (to be pointed out at the end). Could someone please address them?
Statement of the theorem: Let $\{M_n\}$ be martingale such that $\sup_n \|M_n\|_2 \leq B < \infty$. Then $M_n$ converges in $L^2$ and almost surely.
Proof: First show that $M_n$ converges in $L^2$. Since $M_n^2$ is a submartingale, $\mathbb{E}[M_n^2]$ is increasing. Put $C = \lim_{n \rightarrow \infty} \mathbb{E}[M_n^2]$. Then \begin{align*} \mathbb{E}[(M_{n + k} - M_n)^2] & = \mathbb{E}[M_{n + k}^2] - \mathbb{E}[M_n^2]\\ & \leq C - \mathbb{E}[M_n^2] \rightarrow 0 \end{align*} as $n\rightarrow \infty$. We now have, $\sup_{k\geq 0} \mathbb{E}[(M_{n + k} - M_n)^2] \rightarrow 0$ as $n\rightarrow \infty$, which shows $\{M_n\}$ is Cauchy in $L^2$, hence convergent.
Next prove a.s. convergence. Put $Z_n = \sup_{i, j \geq n} |M_i - M_j|$. Then $$ E[Z_n^2] \leq 2 \lim_{N \rightarrow \infty} \mathbb{E}[\max_{n \leq k \leq N} |M_k - M_n|^2] $$
Apply the following version of Doob's inequality (which says for a positive submartingale $X_n \in L^2$, we have $\mathbb{E}[(\max_{1 \leq k \leq n} X_k)^2] \leq 4 \mathbb{E}[X_n^2]$) to the martingale $\{(M_{n + k} - M_n)\}_{k = 0}^\infty$, we find $\mathbb{E}[\max_{n \leq k \leq N} |M_k - M_n|^2] \leq 4 \mathbb{E}[|M_N - M_n|^2]$. This gives us $$\mathbb{E}[Z_n^2] \leq 8 \lim_{N \rightarrow \infty} \mathbb{E}[|M_N - M_n|^2]$$ Since we have shown that $M_n$ converges in $L^2$, taking $n \rightarrow \infty$ in the above inequality gives $\lim_{n \rightarrow \infty} E[Z_n^2] = 0$. Since $Z_n$ is decreasing, $Z = \lim_{n \rightarrow \infty} Z_n$ exists a.s., and by Fatou's lemma, $\mathbb{E}[Z^2] = 0$. We find $Z = 0$ a.s., from which we conclude that $M_n$ is a.s. Cauchy. This finishes the proof.
I understand the main idea of the proof. Here are the three details that I don't follow:
How was the inequality $\mathbb{E}[Z_n^2] \leq 2 \lim_{N \rightarrow \infty} \mathbb{E}[\max_{n \leq k \leq N}|M_k - M_n|^2]$ established? I know that for $n \leq i, j \leq N$, $|M_i - M_j| \leq |M_i - M_n| + |M_j - M_n|$, so $|M_i - M_j|^2 \leq 4 \max_{n \leq k \leq N} |M_k - M_n|^2$. Taking expectations, we get $\mathbb{E}[\max_{n \leq i, j \leq N}|M_i - M_j|^2] \leq 4 \mathbb{E}[\max_{n \leq k \leq N} |M_k - M_n|^2]$. By Fatou's lemma, \begin{align*} \mathbb{E}[Z_n^2] & \leq \liminf_{N \rightarrow \infty} \mathbb{E}[\max_{n \leq i, j \leq N} |M_i - M_j|^2]\\ & \leq 4 \lim_{N \rightarrow \infty} \mathbb{E}[\max_{n \leq k \leq N}|M_k - M_n|^2] \end{align*} This is as close as I get - notice the factor of $4$. How did the original proof end up with a factor of $2$?
Why does the limit $\lim_{N \rightarrow \infty} \mathbb{E}[|M_N - M_n|^2]$ exist?
Why does the double limit $\lim_{n\rightarrow \infty} \lim_{N \rightarrow \infty} \mathbb{E}[|M_N - M_n|^2]$ exist and equal zero? This should follow from the fact that $M_n$ is Cauchy in $L^2$, but I cannot find a precise argument.