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## Homework Statement

Given a Cauchy sequence of integers, prove that the sequence is eventually constant.

**2. Relevant Definitions and Theorems**

Definition of Cauchy sequences and convergence

Monotone convergence

Every convergent sequence is bounded

Anything relevant to integers

I can see why the theorem is true. I thought that, since the sequence in nonempty and bounded, the supremum and infimum of the set containing the sequence both exist and that the limit of the sequence must be a number between the infimum and the supremum. But I got stuck trying to prove that the limit is contained within that set (that the limit is also an integer). I don't know if it's my approach that's leading me to a dead end or if there's a theorm I've overlooked or something else entirely. Maybe I'm making it overly complicated? Any help would be appreciated.

Definition of Cauchy sequences and convergence

Monotone convergence

Every convergent sequence is bounded

Anything relevant to integers

## The Attempt at a Solution

I can see why the theorem is true. I thought that, since the sequence in nonempty and bounded, the supremum and infimum of the set containing the sequence both exist and that the limit of the sequence must be a number between the infimum and the supremum. But I got stuck trying to prove that the limit is contained within that set (that the limit is also an integer). I don't know if it's my approach that's leading me to a dead end or if there's a theorm I've overlooked or something else entirely. Maybe I'm making it overly complicated? Any help would be appreciated.