Basic Analysis I — Real Analysis · Problem 2.5.12

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · cauchy-sequences-series

Problem

Let xn:=∑i=1n1/ix_n := \sum_{i=1}^{n} 1/i. Show that for every kk, we get lim⁡n→∞∣xn+k−xn∣=0\lim_{n \to \infty} |x_{n+k} - x_n| = 0, yet {xn}n=1∞\{x_n\}_{n=1}^{\infty} is not Cauchy.

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