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($S$).