In: Advanced Math
Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, am > an for all n > m
The solution is given below. In part a, the subsequence of crests is a monotonically decreasing subsequence. The infimum exists as the given sequence is bounded. This infimum is the limit. In part b, the subsequence that we get will be a monotonically increasing subsequence and the supremum will be the limit.