Extended real subsequence theorem
= Extended real subsequence theorem
Every real <sequence> has a <subsequence> converging to an <extended real number>. A bounded sequence has a real convergent subsequence by the <Bolzano-Weierstrass theorem>. If it is unbounded above, every tail is unbounded above, so choose increasing indices $n_j$ with $a_{n_j}>j$. These terms tend to positive infinity. If it is unbounded below, use $a_{n_j}<-j$.