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 $a_{n_j}<-j$.