# Non-increasing Monotone Sequence Convergence Proof

My goal is to prove the monotone convergence of a non-increasing sequence of real numbers. There are some steps in the proof that I'm not sure about. The question:

If $S$ is a non-increasing sequence bounded below, show that it converges.

Here is what I have so far:


Since S is non-increasing and bounded below, by the Greatest Lower Bound property, it has an inf.

Let $c$ = inf($S$). Goal: Show that $S$ converges to $c$.

By definition of infimum, for every $\epsilon$ $> 0$, $\exists N$ such that $S_N$ $< c + \epsilon$ (Otherwise, $c + \epsilon$ would be a lower bound of $S$ and that contradicts $c$ being the infimum of $S$.

Here is where I am unsure about my proof:


Because $S$ is non-increasing, if $n > N$, $\epsilon > \mid S_n - c \mid \geq \mid S_N - c\mid$.

So, that proves that $S$ converges to $c$, which is inf($S$).

You are correct for the most part. You just have to be careful about the final inequalities -- switch $n$ and $N$.

Since the sequence is non-increasing, if $n \geq N$ then $S_n \leq S_N$, and

$c \leq S_n < c + \epsilon$.

So

$0 \leq S_n-c < \epsilon$

and

$|S_n - c| < \epsilon$.

• Thanks. Now that I think about it, it makes sense! – user3025403 May 10 '14 at 21:26