True or False? There exists a martigale $\left(M_{n}\right)_{n}$ such that $\mathbb{P}\left(\left|M_{n}-M_{n-1}\right| \leq 10\right)=1 \quad \forall n$ and $\mathbb{P}\left(\lim _{n \rightarrow \infty}\left|M_{n}\right|=+\infty\right)=1$
My attempt
I thought of using dominating convergence theorem to prove this statement is false.
$\mathbb{P}\left(\left|M_{n}-M_{n-1}\right| \leq 10\right)$ implies that $E\left[\left|M_{n}-M_{n-1}\right|\right] \leq 10$
Hence, by Jensen's inequality we have: $E\left[M_{n}\right] \leqslant E\left[\left|M_{n}\right|\right] \leqslant 10+E\left[|M_{0}|\right]$
Thus by dominating convergence theorem: $$ \lim _{n \rightarrow \infty} E\left[M_{n}\right]=E\left[M_{\infty}\right]<10+E\left[|M_{0}|\right]$$
Hence $M_n$ converges to $M_{\infty}$ a.s and this statement is false.
I am not sure about using dominating convergence theorem as the theorem requires $M_n$ to converge in order to use. Thank you for your help!
A martingale with bounded jumps, like $(M_n)$, has the following property (see Durrett's Probability: Theory and Examples, https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf, Theorem 4.3.1): If $C:=\{\omega: \lim_nM_n(\omega)$ exists in $\Bbb R\}$ and $D:=\{\omega:\limsup_n M_n(\omega)=+\infty,\liminf_n M_n(\omega)=-\infty\}$, then $\Bbb P[C\cup D]=1$.
Now ask yourself how $\{\omega:\lim_n|M_n(\omega)|=+\infty\}$ is related to $C$ and $D$.