Hey I'm new here. I need help with Intro to Real Analysis;completely lost

Hey I'm new here.

I'm going to college to become a high school math teacher, but some of these math courses are pretty intense. Actually the only ones that are difficult are the proof based courses. I really need help.

This is my homework and I'm completely lost and unsure how to go about doing it. We started and completely went over this topic for supS and infS in our last class. That was last Thursday. We are on spring break so we have time, but it's due next Tuesday. My teacher said he would be unavailable over break for questions so I'm SoL. Help would be greatly appreciated. I hope I'm posting this in the right section of the forums. This is for Real Analysis.

1. For each set, ﬁnd inf S and sup S. State whether or not these numbers are in S.

(a) S ={x : x^2 − 3 < 0}

(b) S ={y : y = (x^2)/(x^2 + 1), x ≥ 0

(c) S = {x : 0 < x < 5, sin x = 0}

2. Let S_1, S_2 be subsets of R1. Deﬁne S = {x : x = x_1 · x_2, x_1 ∈ S_1, x_2 ∈ S_2}. Find sup S and inf S in

terms of sup S_1, inf S_1, sup S_2, and inf S_2. Discuss all cases.

3. Let S be a nonempty set of real numbers that is bounded below. Let L = inf S and suppose that

L is not an element of S.

(a) Prove that every open interval containing L contains inﬁnitely many elements of S.

(b) Prove that there is a nonincreasing sequence {x_n} such that x_n ∈ S for all n ∈ N and

lim n→∞ x_n = L.

Re: Hey I'm new here. I need help with Intro to Real Analysis;completely lost

Quote:

Originally Posted by

**vinmanvji** Hey I'm new here.

I'm going to college to become a high school math teacher, but some of these math courses are pretty intense. Actually the only ones that are difficult are the proof based courses. I really need help.

This is my homework and I'm completely lost and unsure how to go about doing it. We started and completely went over this topic for supS and infS in our last class. That was last Thursday. We are on spring break so we have time, but it's due next Tuesday. My teacher said he would be unavailable over break for questions so I'm SoL. Help would be greatly appreciated. I hope I'm posting this in the right section of the forums. This is for Real Analysis.

1. For each set, ﬁnd inf S and sup S. State whether or not these numbers are in S.

(a) S ={x : x^2 − 3 < 0}

(b) S ={y : y = (x^2)/(x^2 + 1), x ≥ 0

(c) S = {x : 0 < x < 5, sin x = 0}

2. Let S_1, S_2 be subsets of R1. Deﬁne S = {x : x = x_1 · x_2, x_1 ∈ S_1, x_2 ∈ S_2}. Find sup S and inf S in

terms of sup S_1, inf S_1, sup S_2, and inf S_2. Discuss all cases.

3. Let S be a nonempty set of real numbers that is bounded below. Let L = inf S and suppose that

L is not an element of S.

(a) Prove that every open interval containing L contains inﬁnitely many elements of S.

(b) Prove that there is a nonincreasing sequence {x_n} such that x_n ∈ S for all n ∈ N and

lim n→∞ x_n = L.

It would help you to remember that the supremum of a function is its least upper bound, and the infemum of a function is its greatest lower bound.

So in the first function, is there a minimum and/or a maximum? From there can you tell if there is a supremum or an infemum?

Re: Hey I'm new here. I need help with Intro to Real Analysis;completely lost

So in 1a, is is saying basically to evaluate the function less than 0. so the infS would be -3 and the supS would be 0, because the boundary is the x axis essentially? If so that makes sense.

I understand supS and infS just not how to go about proving it. My professor showed something with induction, but I just can't seem to figure it out.

I'm also unsure about how to do 1c, and what it's even saying. Some elaboration would be helpful.

Re: Hey I'm new here. I need help with Intro to Real Analysis;completely lost

So in 1a, is is saying basically to evaluate the function less than 0. so the infS would be -3 and the supS would be 0, because the boundary is the x axis essentially? If so that makes sense.

I understand supS and infS just not how to go about proving it. My professor showed something with induction, but I just can't seem to figure it out.

I'm also unsure about how to do 1c, and what it's even saying. Some elaboration would be helpful.......