Recent content by jmjlt88

This book bridges the gap from kindergarten mathematics to Rudin exceedingly well. It is written in a very conversational style- something that many books at this level lack. The author actually wants you to learn, as opposed to Rudin who just throws theorems at you. The exercises are at the...

The contrapositive of the claim would be: "If for every sequence (xn) that converges to c, the sequence (f(xn)) converges, then f has a limit at c." I tried the contrapositive (and also had some difficulty). However, I gave up rather quickly since I am so focus on constructing that sequence...

To pasmith, no, f is not assumed to be continuous. To verty, indeed. A couple ways the sequence (f(xn)) could fail to converge is if we have some asymptotic behavior or something like the signum function. Unfortunately, the hint provided is not getting the wheels in motion. :(

Let A be a subset of ℝ. Let c be a limit point of A. Consider the function f: A → ℝ Claim: If the function f has not have a limit at c, then there exists a sequence (xn), where xn≠c for all n, such that lim xn=c, but the sequence (f(xn)) does not converge. Since the function f does not have a...

Wait... I think I got it.. Pick any y' and x'. Then, g(y') ≤ h(x,y') for all x; therefore g(y')≤h(x',y'). Now, h(x',y')≤sup{h(x',y) : y in Y}=f(x'). This established the following inequality: g(y')≤f(x'). Because x' and y' were arbitrary, we conclude g(y)≤f(x) for all x,y. Thus g(y) is...

Let X and Y be nonempty sets and let h: X x Y→ℝ Define f: X →ℝ and g: Y→ℝ by the following: f(x)=sup{h(x,y): y in Y} and g(y) = inf{h(x,y): x in X} Prove that sup{g(y): y in Y} ≤ inf{f(x): x in X} Attempt at solution: Pick y' in Y. Then g(y')≤h(x,y') for all x in X. Hence, there...

Let me add a remark. By "easiest textbook," I mean that the author(s) is/are actually trying to teach you. For instance, I enjoyed Munkres' very expository style in his Topology text. The aforementioned textbook read as though he was carefully explaining the material to you. It was a pleasant...

The textbook of which I am interested in obtaining some input are the following: Cohn- Measure Theory Wheeden and Zygmund - Measure and Integral Folland- Real Analysis. Let me be frank here, I am looking for the easiest textbook. I will be working through the text for a self-study...

Greetings! Can anyone tell me a little bit about the book Integral, Measure and Derivative: A Unified Approach by Shilov? Is it suitable for self-study? I am wishing to study the basics of measure theory. I will be using the text alongside Kreyszig's Funcational Analysis. Having already...

Right! Now, because M1<x≤M we have that ______________ .

"We can also write ϵx≤sup(B) . Alsoϵx≤ϵsup(A) also. We may have ϵsup(A)≤sup(B) or sup(B)≤ϵsup(A) in which case the equality holds." I am not sure what you are getting at in the above statement. Read carefully the definition of the supremum of a subset of the real numbers. In my previous...

"The supremum of A is an element of A..." The supremum of A may or may not be an element of A. For example, what is the sup A if A=(1,2)? What about if A=[1,2]? If we let, say, M= sup A, then one part of our definition says that x≤M for all x in A. Since ε is a small positive number, εx≤εM for...

WannabeNewton, yes, you are right! I should have said that there are no downsides to the book when used as a supplement. For those interested diving deep into Set Theory (and perhaps never surfacing), Jech's tome would certainly be one route to take. =)

Pinter's text served as my gateway back into rigorous mathematics! It is well-written and organized. There are a plethora of perfecty chosen problems at the end of each chapter. I highly recommend this text to anyone. In fact, I will be suggesting this text to one of my brighter students for...

I skimmed this book while reading the opening chapter in the book Topology by James Munkres. The writting style of Paul Halmos is superb. There are not many problems designated as exercises; however, as I believe the author points out, the entire book is essentially an exercise. Hence, you will...