 Reciprocal limits and escape in absolute value

For a real sequence with no zero terms, the displayed equivalence follows directly by comparing $|1/a_n|$ with $1/M$. Escape to positive infinity implies reciprocal convergence to zero, but the converse does not retain the sign: $a_n=-n$ is a counterexample. If $a_n\to a\ne 0$, eventually $|a_n|>|a|/2$, and $|1/a_n-1/a|\le 2|a_n-a|/|a{|}^2$. This proves reciprocal convergence at a nonzero finite limit of a sequence.