= Reciprocal limits and escape in absolute value
{title2=$1/a_n\to0\ \Longleftrightarrow\ |a_n|\to\infty$}
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 $a_n\to a\ne0$, eventually $|a_n|>|a|/2$, and $|1/a_n-1/a|\le2|a_n-a|/|a|^2$. This proves reciprocal convergence at a nonzero finite <limit of a sequence>.
Back to article page
