# Thread: Set, monotone sequence, inf

1. ## Set, monotone sequence, inf

Hi, I have a question about the following proposition:

Let S R be a nonempty bounded set. Then there exists a monotone sequence {xn} such that xn S for each n N and limxn = inf (S).

How would one go about proving this? I know that a monotone sequence means that xn is greater or equal to xn+1 (or less than or equal to, but in this situation, because we want inf(S), we want monotone decreasing, right?).

Help?

2. Originally Posted by seven.j
Hi, I have a question about the following proposition:

Let S R be a nonempty bounded set. Then there exists a monotone sequence {xn} such that xn S for each n N and limxn = inf (S).

How would one go about proving this? I know that a monotone sequence means that xn is greater or equal to xn+1 (or less than or equal to, but in this situation, because we want inf(S), we want monotone decreasing, right?).

Help?
If $\inf S\in S$ choose the sequence to be the constant sequence $x_n=\inf S$, otherwise $\inf S$ is a limit point of $S$. So, try choosing arbitrary neighborhoods of $S$ and choosing arbitrary points in them.