Michigan State University Real Numbers and Bounded Sequence Worksheet need to do it before 3:50. and finish it as a pdf file. please write as tidy as you c

Michigan State University Real Numbers and Bounded Sequence Worksheet need to do it before 3:50. and finish it as a pdf file. please write as tidy as you can. 1. (a) Let A CR be a non-empty set that is bounded from above. Define sup A.
Say if the following statements are true or false. If true, give a short proof; if false, give a counterexample.
(b) If A, B CR are nonempty sets that are bounded from above and An B is nonempty, then
sup(ANB) = min{sup A, sup B).
(c) If A, B CR are nonempty sets that are bounded from above, then
sup(AUB) = max{sup(A), sup(B)}.
(d) If A is non-empty and bounded from above, then there exists a sequence of real numbers (an) such that
(an) + sup A as n goes to infinity, where are A for all n e N.
(e) Let (In), (yn), (2n) be sequences of real numbers such that, for some N EN, Yn In 32 for all n > N.
If yn) and (2) both converge, then (In) has a convergent subsequence.
2. (a) Let (In) be a sequence. Define precisely what it means for (In) to converge to IER.
(b) Using only the definition from part (a) show that
3n – 4
non + 2
lim
= 3
(c) Does (-1)”
3n-4
n+2
converge? Justify your answer.
2.2K/s
3. Let a be a bounded sequence.
(a) Define lim supr. 4, and lim inf a +
nh
(b) Let an = sin
Compute lim supr+ , and lim infr
En + r. (Justify your answers)
1
n
(c) Let br · sin (“,). Compute lim supn+ by and lim infa+ae by(Justify your answers)
4. Let (ar) be a sequence of real numbers such that a1 = { and an+1 = /20n + 15. Show that the sequence is
increasing, that an