Hi, I am in difficulty when I kept trying to do these questions.

Could you give me a help to solve these questions?

Thank you so much.

Note: I attach the extra information as well

Q1 (7 points) 4-l-1 (a) Let 3,, : ant\02 (5 points) Let so = 1 and 31 = 1. Forn 2 2, let 3\Q3 (4 points) Let A C R be bounded, and let a = inf A. Prove that there exists a sequence (@n ) with each En E A such that n -> a. (One approach is to describe how to choose each , making sure that each such choice is well defined.)Q4 (6 points) Prove that the following statements are equivalent for any nonempty set A C R. (Two or more statements are called "equivalent" if each one of them implies all of the others. Here there are only two statements, so saying they are equivalent is the same as saying "(1) if and only if {2}." (1) A is dense in R. {That is, for any a, b E IR with a, 0 and note we need to show |Ksn - ks| N implies sn - s| N implies |ksn - ks 0; we need to show Isn + tn - (s + t)| N1 implies |sn - s) . Likewise, there exists N2 such that n > N2 implies Itn - t) . Let N = max{ N1, N2}. Then clearly n>N implies Isnttn-(stt)| 0. By Example 6 of $8, there exists m > 0 such that ( sn| 2 m for all n. Since lim sn = s there exists N such that n > N implies |s - Sn| N implies 1 Is - Sul. s - sul 0. (b) limn too an = 0 if la| 0. 0 Proof (a) Let E > 0 and let N = (1)1/P. Then n > N implies n? > and hence e > mp. Since mp > 0, this shows n > N implies Imp - O| 0. By