Basic Analysis I — Real Analysis · Problem 6.2.14

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · double-sequence-joint-limit

Problem

A joint limit does not mean the iterated limits exist. Consider xn,m:=(−1)n+mmin⁡{n,m}x_{n,m} \mathrel{:=} \dfrac{(-1)^{n+m}}{\min\{n,m\}}. (a)) Show that for no nn does lim⁡m→∞xn,m\lim_{m \to \infty} x_{n,m} exist, and for no mm does lim⁡n→∞xn,m\lim_{n \to \infty} x_{n,m} exist. So neither lim⁡n→∞lim⁡m→∞xn,m\lim_{n \to \infty} \lim_{m \to \infty} x_{n,m} nor lim⁡m→∞lim⁡n→∞xn,m\lim_{m \to \infty} \lim_{n \to \infty} x_{n,m} makes any sense at all. (b)) Show that the joint limit of {xn,m}n,m=1∞\{x_{n,m}\}_{n,m=1}^\infty exists and equals 00.

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