1. Apply Definition 2 from Section 4 (i.e., the precise definition of limit) to prove that the following sequences converge. (Hint: You will need to first determine where they converge to.) (a) lm...

1. Apply Definition 2 from Section 4 (i.e., the precise definition of limit) to prove that the following sequences converge. (Hint: You will need to first determine where they converge to.) (a) lm ni,co "1
(b) urn 2"—P 00 n +1 (c) thin4n+1 n—too 271+5
(d) lim (-/F 1 — %/it) n—too
2. (a) Use the precise definition of limit to prove that if {xn} converges to x, then {Ixnl} converges to Ix'. (b) Give an example to show that the converse is not true. 3. Prove that if {xn} converges to 0 and {yn} is a bounded sequence, then {xnyn} converges to
0.
4. (Bonus!) Give an example of two divergent sequences {xn} and {yn} such that the sequence {xnyn} converges.