Prove a stronger version of Proposition 2.3.7. Suppose {xn}n=1∞ is a sequence such that every subsequence {xnm}m=1∞ has a subsequence {xnmi}i=1∞ that converges to x.
a) First show that {xn}n=1∞ is bounded.
b) Now show that {xn}n=1∞ converges to x.