How does the sandwich argument for sequence convergence work?

Question

I'm taking a course in probability and the professor is talking about convergence. He said:

If $|a_i - a | \leq b_i$, for all $i$, and $b_i \to 0$, then $a_i \to a$.

He said there is a sandwich argument which says if we have one number that converges to some number A and another sequence which converges to that same number A and our sequence is somewhere in between, then our sequence must also converge to that particular number A.

I'm still not understanding it, especially the formula.

Answer 1

Since $0 \leq |a_i - a |\leq b_i$ then taking limits we have $0 \leq \lim_{i\to\infty} |a_i - a |\leq 0$ so the limit in the middle must also be zero and we can conclude that $a_i \to a$