(a) Let be a bounded sequence of real numbers. Show that has a convergent subsequence.
(b) Let be a bounded sequence of complex numbers. For each , write . Show that has a subsequence such that converges. Hence, or otherwise, show that has a convergent subsequence.
(c) Write for the set of positive integers. Let be a positive real number, and for each , let be a sequence of real numbers with for all . By induction on or otherwise, show that there exist sequences of positive integers with the following properties:
for all , the sequence is strictly increasing;
for all is a subsequence of and
for all and all with , the sequence
Hence, or otherwise, show that there exists a strictly increasing sequence of positive integers such that for all the sequence converges.