Basic Analysis I — Real Analysis · Problem 6.2.13

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · double-sequence-iterated-limits

Problem

Suppose the joint limit of {xn,m}n,m=1∞\{x_{n,m}\}_{n,m=1}^\infty is LL, and suppose that for all nn, lim⁡m→∞xn,m\lim_{m \to \infty} x_{n,m} exists, and for all mm, lim⁡n→∞xn,m\lim_{n \to \infty} x_{n,m} exists. Then show lim⁡n→∞lim⁡m→∞xn,m=lim⁡m→∞lim⁡n→∞xn,m=L\lim_{n \to \infty} \lim_{m \to \infty} x_{n,m} = \lim_{m \to \infty} \lim_{n \to \infty} x_{n,m} = L.

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