I've got a question regarding a step in a proof, the situation is following:
Let $X_{1},\dots,X_{n}$ be independent, symmetric stochastic variables so that $\sum\limits_{n=1}^{\infty}X_{n}$ exists in probability. If I use the fact that convergent in probability implies the same convergence of the corresponding Cauchy sequence i can write: $\lim_{n,m\to\infty}P(|S_{m}-S_{n}|>t)=0$ where $S_{n}=\sum\limits_{i=1}^{n}X_{i}$. My textbook then says that from the triangle inequality we can write that as: $\lim_{n\to\infty}\sup_{m\geq n+1}P(|S_{m}-S_{n}|>t)=0$ Could someone please help me fill in the blanks?
I hope I didn't leave anything relevant out, let me know if you feel i did.