We work in the real numbers. Are the following true or false? Give a proof or counterexample.
(a) If converges, then converges.
(b) If converges, then converges.
(c) If for all , and converges, then as .
(d) If , for all , and converges, then as .
(e) If is a decreasing sequence of positive numbers, and converges, then as .
(a) and (b) can be proved similarly. Since converges, for some , when , then . Take s.t., . That is, . Also, . This implies that and therefore converges.
(c) This is where I get confused. THis seems like an indeterminate form, we have never done this in class. same for (d) and (e). any suggestions?