Jeremy Kun

I'm sorry to hear you're struggling with the exercises. I know it can be frustrating. For what it's worth, I don't think you need to solve every exercise, and certainly...

Definition 4.3 defines the notation, but I agree it's confusing juxtaposed with the previous exercise. If you submit an erratum at pimbook.org, I'll fix it and credit you :) Thanks...

I should have instead wrote the question as: Suppose for each natural number n we chose a *different, countably infinite* set $A_n$. I.e., each $A_n$ has a bijection with the...

Sorry, I was mistaken. The previous exercise already uses that concrete construction. I suppose the point of this exercise was to show that N x N is not special in...

It is exactly the Lagrange interpolating polynomial, though I didn't hear that term when I first saw this. To the best of my memory, I originally discovered this way of...

@abjrcode Those two are the same, and you can see this if you multiply -1 on the top and bottom of each term. Also, if you suspect whether they're the...

This one requires some more elbow-grease than it appears at first glance. I think the best vague hint I can give is to pose the question: how do you know...

I think you have the right idea, but being a little bit less direct will help keep the reasoning more focused on the general proof. When I think about sqrt(2)...

I misspoke above about "reducing any power", but hopefully you still get the idea? I am trying to convey the idea that you can get from large powers of (sqrt(a)...

You're very close, and I think what matters more than rigor is that you understand the core idea. However, in what you wrote a^n and b^n are not necessarily rational....