Bounded sequence which is not convergent, but differences of consecutive terms converge to zero

real-analysis sequences-and-series examples-counterexamples

A single example will do the job. Try this: $$0,1,1/2,0,1/3,2/3,1,3/4,2/4,1/4,0,1/5,2/5,3/5,4/5,1,5/6,4/6,3/6,2/6,1/6,0,1/7,\dots.$$

We travel back and forth between $0$ and $1$, first with a giant step, then with two steps of length $1/2$, then with $3$ steps of length $1/3$, and so on. Note that every real number between $0$ and $1$ is a limit of a subsequence of our sequence.


They're asking for a sequence where successive terms get closer to each other but don't converge on a final value. An example that comes to mind would be $\sin(\sqrt{n})$. The gap between successive values of $\sqrt{n}$ gets smaller, but obviously the series is going to oscillate.

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