Convergence and Cauchy Sequences in Rational Numbers

[vertigo74]

Homework Statement

Prove that if {a_{n}} is a sequence of rational numbers such that {a_{n+1}} > {a_{n}} for all n \in \textbf{N} and there exists an M\in \textbf{Q} such that {a_{n}} \leq M for all n \in \textbf{N}, then {a_{n}} is a Cauchy sequence of rational numbers.

Homework Equations

Do not use the least upper bound property.

A sequence is Cauchy in the rational numbers if \exists an N \in\textbf{N}, such that |{a_{n}} - {a_{m}} | < \epsilon for all n, m \geq N.

If a sequence converges, it is Cauchy.

The Attempt at a Solution

I understand why this is true, but I am having trouble formulating the math to do a proof behind it. I can see that if the sequence never gets bigger than M, and that it is strictly increasing, the sequence must start converging and be Cauchy, but I'm kind of confused at how to start doing the epsilon stuff.

Thanks!

Edit: I'm not sure why those are showing up as superscripts. They are supposed to be subscripts.

 

[vertigo74]

someone please heeeeeeelp

 

[dalle]

try a proof by contradiction.
suppose a_n is no cauchy sequence.For a fixed \epsilon &gt; 0show there is a subsequence b_n of a_n such that
b_{n+1} &gt; b_n + \epsilon