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## Homework Statement

Say {p_n} is a sequence in R, abs(p_n-p_n+1) -> 0, and {p_n} is bounded. Is it true that {p_n} must converge?

## Homework Equations

## The Attempt at a Solution

Intuition: Yes; in my attempts to find a counterexample I found sequences which diverged even though successive terms were closer (Partial sums of the Harmonic series), and were not bounded. From diagramming it seems to me that if you have the range of p_n bounded above and below, and the distance between successive terms must decrease, then the boundary will continue to close in from both sides until the sequence converges.

The only thing is I'm having trouble proving this without the assumption that the sequence is monotonic (I can't get a contradiction out of the existence of sup p_n or inf p_n)

So I'd like to make sure, before I continue with the proving, that this is actually true.

EDIT: I actually think this is false. Don't have a counter-example yet, but I think you might could have like, a sequence defined which oscillates around two points... I'm just not sure if I can do that without the oscillation being a convergent one.

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