Let $\lbrace X_n\rbrace _{n=1}^\infty$ be the series of independent random variables and $S_n=X_1+X_2+...+X_n.$ Prove that if $\frac{S_n}{n}$ converges in probability towards $0$, then $\frac{X_n}{n}$ converges towards $0$ in probability. I was instructed to use $\frac {X_n} n= \frac {S_n} n- \frac {n-1} n \frac {S_{n-1}} {n-1}$ to prove it, but I can't seem to be able to figure out how to use it in the definition of convergence in probability.
2026-04-07 10:06:00.1775556360
Proof regarding convergence in probability.
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