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

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.

• Think of $x_{n+1} - x_{n}$ as the derivative of $x_n$. Does a function with a derivative approaching zero need to be bounded above? – newToProgramming Oct 14 '13 at 2:41
• – Martin Sleziak Dec 3 '16 at 7:31

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