Basic Analysis I — Real Analysis · Problem 2.2.3

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · continuity-of-algebraic-operations

Problem

Prove that if

\{x_n\}_{n=1}^\infty

is a convergent sequence,

k \in \mathbb{N}

, then

\lim_{n\to\infty} x_n^k = \left(\lim_{n\to\infty} x_n\right)^k.

Hint: Use induction.