Cauchy sequences and condition $X_{n+1} – X_n\to 0$

Question

Let $\{X_n\}$ be a sequence and suppose that the sequence $\{X_{n+1} – X_n\}$ converges to $0$. Give an example to show that the sequence $\{X_n\}$ may not converge. Hence, the condition that $|X_n-X_m| < \epsilon$ for all $m,n \ge N$ is crucial in the definition of a Cauchy sequence.

Answer 1

Hint: What can you say about the partial sums of, say, the harmonic series?

Or a sequence involving $\ln{n}$?

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More generally, think of your favorite function $f$ with $\lim_{n \to \infty} f(n) = \infty$, but that approaches $\infty$ "slowly."

Answer 2

Take the sequence $X_n = \sum_1^n \frac{1}{n}$. Here $X_{n=1} - X_n = \frac{1}{n}$ which goes to $0$ but $X_n$ is not convergent.