Transcribed Text
1. (20) For each of the following subsets of ℝ, give its maximum, minimum, infimum,
and supremum, if they exist. Otherwise, write "none".
(a) {1,4} (b) [0,4] (c) {
} (d) {
}
(e){
(
) } (f) ⋂ (
)
(g) ⋃ [
]
(h) (i)
(j) {∑
2. (10) Prove that every sequence of real numbers has a monotone subsequence.
3. (10) The sequence given by
converges. Find its limit.
4. (10) Prove √
.
5. (10) Prove that if , then √
√
for all .
6. (10) For any sequence
, let
(a) Prove that implies .
(b) Give an example of a divergent sequence such that
converges.
7. (10) Suppose that (an) is a sequence of numbers such that for all n, 

where ∑ is convergent. Show that (an) converges.
8. (20) Prove the following, using Definition 2.2.3(convergence of a sequence):
(a)
(b)
These solutions may offer stepbystep problemsolving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skillbuilding and practice.
Unethical use is strictly forbidden.