Solution
= Solution
The general principle of convergence says that a real <sequence> converges exactly when it is Cauchy. Convergent <sequences> are Cauchy by the triangle inequality. Conversely, a Cauchy <sequence> is bounded, so Bolzano–Weierstrass gives a convergent subsequence $a_{n_k}\to a$; the Cauchy property then forces the entire <sequence> to converge to $a$.