Here is my question and my solutions. I am just curious if this makes sense.
If something is wrong, then it will be nice if you can correct it for me or provide another solution. Please and thank you!
2026-03-29 15:02:30.1774796550
Showing that the absolute value of |X| converges almost surely.
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To elaborate on copper.hats comment, if $\mathbb P(X>10)>0$ then there exists some integer $m$ such that $\mathbb P\left(X>10+\frac1m\right)$. But this implies that $\mathbb P\left(|X-X_m|\geqslant \frac1m\right)$, and contradicting the assumption that $$ \lim_{n\to\infty}P\left(|X-X_n|\geqslant \frac1n\right) = 0. $$