Does a monotone increasing sequence in a bounded set always converge to the supremum?

[mathanon]

Let A be a nonempty subset of R that is bounded above and let α=supA. Show that there exists a monotone increasing sequence {an} in A such that α=lim an. Can the sequence {an} be chosen to be strictly increasing?

 

[HallsofIvy]

For every positive n, look at the set $\{ a_i| |a_i- a|<1/n\}$. Can you see that this set is non-empty for all n? Choose a member of this set to be in the subsequence.