Recent content by cmurphy

Would it be at all helpful to look at the fact that lim inf sn = -lim sup -sn, or is that just complicating matters? Colleen

HallsofIvy, what are you supposing is not true? Do you follow my proof and then suppose that Case 2 is not true? Or are you supposing the original statement is not true? I understand where you are going ... that your assumption would show that a + epsilon would then become the infimum...

Yes, the problem does say that 0 < sn <2, and that sn+1 = root(sn +2). Thus I said that (sn+1)^2 = sn + 2. Then sn = (sn+1)^2 - 2 Thus 0 < sn = (sn+1)^2 - 2 < 2 And 2 < (sn+1)^2 < 4 So root 2 < sn+1 < 2. Thus I have that sn+1 < 2. I obviously have that 0 < sn, from the given...

Also, the fact that lim sup sn < s for some real number s is given. Then I just made the substitution lim sup sn = lim n->infinity sup{sn: n>N}

The definition in our books is that: lim sup {sn: n>N} n->inf i.e. For large n, the lim sup sn is the limit of all of the suprema. They also defined lim sup sn to be exactly the supremum of the set of subsequential limits.

I need to prove that the limit of a constant sequence converges, using the definition of a limit. This is what I have: Let e > 0 be given. Then |sn - s| < e But sn = s for all sn, thus |s - s| < e |0| < e 0 < e Thus N can be any number? This proof is simple, but I am making it...

Let (sn) be a real sequence and s in R. Prove that lim sup (sn) < s implies sn < s for n large. My answer seems too easy. Is there anything missing? Given lim sup (sn) < s. By definition of lim sup, we know lim N->infinity sup {sn: n > N} < s Then for n > N, we must have sn < s...

Problem: Suppose 0 < sn < 2 and sn+1 = root (sn + 2) for n in N. Prove 0 < sn < sn+1 < 2 holds for all n in N. Does sn converge? If so, what is the limit. I am able to show that sn+1 < 2 by squaring the equation sn+1 = root (sn + 2) and making a substitution. How would I go about...

Another question: Prove that if a set E in R has a finite infimum and e > 0 is any positive number, then there is a point a in E such that inf E <= a < inf E + e. The first part, inf E <= a, is obvious from the definition of infimum. I am having trouble showing that a < inf E + e, even...

Hello, I am taking Adv. Calc, and we have a test next week. I am going to post a few questions that I have from the review where I got stuck. If you have any help, please steer me in the right direction! Question 1: Suppose sn <= 0 <= tn for n in N. Prove (lim inf sn)(lim sup tn) <=...

lim sup is the limit of the supremum for large N, right? lim sup sn = lim as N-> infinity of the set {sn : n > N}. So begin listing some terms of sn: 1, 1/2, 1/3, 1/4, 1/5, 1/6, ... If N = 1, then n must be at least 2. So we ignore the first term, and look at the sequence 1/2, 1/3, 1/4...

I understand what you've done, but I don't actually see how to proceed. Since sn + tn - 2e <= lim sup sn + lim sup tn, how do you get lim sup (sn + tn)? I think this might go back to the problem that I'm having with understanding how to manipulate lim sup. Conceptually, I understand what it...

Hello, I have several questions regarding lim sups and lim infs. I have a couple of proofs that I need to do, and I'm not sure where to start, because I don't have a good understanding of how to "play" with the definition; lim sup sn = lim N -> infinity sup{sn: n > N}. Any suggestions...

I need to show that the absolute value of x = max {x, -x}. Of course I know this is true, but I must show this. How would I go about doing such a thing?

How would I go about doing a proof by contradiction? Suppose that (a, b) is dense. Also suppose that there are a finite number of elements of E in (a, b). To me this doesn't seem easier. Where would I go from here? Also, I know that (1, 2) intersect (3, 4) would not be dense, but that...