I found this proof on the Internet, but from what I know about convergence in probability, it seems like a $\lim $ would be enough.
Am I right, or am I missing something?
2026-04-03 01:52:58.1775181178
Why does this proof about random variable convergence in probability contain $\limsup$ and not $\lim$?
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Well, the last two lines prove that the $\lim\limits_{n\to\infty}$ of all those quantities actually exists and it is $0$. They used $\limsup\limits_{n\to\infty}$ because they had not proved its existence yet.
$\limsup\limits_{n\to\infty}$ has not only the advantage of preserving inequalities, but it always exists. So in many cases it is the wisest and most accurate choice.
Edit: After giving a better look, technically the doubt is on the existence of $\lim\limits_{n\to\infty}P(|X_n+Y_n-X-Y|\ge\varepsilon)$. The author must put $\limsup$ there. Therefore, all the successive upper bounds should be with $\limsup$. The last one is actually an existing limit by hypothesis.