# Search results

1. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

This is great! I can use this as a reference for future problems as well, thank you!
2. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

Honestly thinking about things in this way really helps I can tell the difference. When I do proof problems from my linear algebra book, it normally doesn't take much time at all because everything seems much more straightforward. The thinking for these kinds of proofs just seems different for...
3. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

No worries! All of this is very helpful. I think I just need to stick it out for the simpler stuff, but also not be afraid to move along when necessary. The thing you said about definitions is something that has tripped me up more than once, no one has ever pointed it out to me before now.
4. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

Thanks for the advice! I don't know why I always feel the need to do every single part of a textbook. Just recently, someone told me that they normally only do half of the problems in a textbook. I had spent weeks doing every problem in every section of the books I'm working on and was wondering...
5. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

No this is like set theory stuff. In the book I'm using, the "real" stuff comes after set theory and properties of the real numbers. Most of the problems are just proofs of definitions similar to this. I've thought about skipping it multiple times but decided against it. I'm starting a degree in...
6. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

Yes, that is the definition I am using. I'll give this approach some thought. Thanks for the response!
7. ### I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

Ok, so here is what I have so far: Suppose ##T_1## is infinite and ##\varphi : T_1 \rightarrow T_2## is a bijection. Reasoning: I'm thinking I would then show that there is a bijection, which would be a contradiction since an infinite set couldn't possibly have a one-to-one correspondence...

Lol
9. ### I Proof about injectivity

Let ##y \in E##. Assume that ##f## is surjective. There is some ##x \in f^{-1}(E)## such that ##y = f(x)##. So by definition, ##x =f(x) \in f(f^{-1}(E))##. At least I finally got something understandable. I didn't have nearly as much trouble proving things about inverse images themselves (i.e...
10. ### I Proof about injectivity

Thank you for this. It took a while to reply because I had to get caught up on some school work. Here's what I worked out: If ##y \in f(f^{-1}(E))##, then ##y = f(x)## for some ##x \in f^{-1}(E)##. So if you have ##x \in f(f^{-1}(E))##, you have it will clearly map back to the set ##E##. So we...
11. ### I Proof about injectivity

Thank you for this. It took a while to reply because I had to get caught up on some school work. Here's what I worked out: If ##y \in E##, then ##y = f(x)## for some ##x \in f^{-1}(E)##. Clearly, ##f^{-1}(E)## is the set of all points that map into ##E##, so ##f(f^{-1}(E))## will give us all...
12. ### I Proof about injectivity

ok, I've thought about it about. The goal of this proof should be to show that ##y \in f(f^{-1}(E) \iff y \in E## So to start with ##y \in f(f^{-1}(E)##. ##y = f(x)## for some ##x \in f^{-1}(E)## Or maybe since it is surjective, it is best to start with ##y \in E## so that we can show that...
13. ### I Proof about injectivity

Ah, ok. I was drawing out a picture and it wasn't making sense.
14. ### I Proof about injectivity

E is a subset of A or B?
15. ### I Proof about injectivity

Wow, I really overcomplicated things XD

Yes!
17. ### I Proof about injectivity

Well if f(x) = f(e) then x = e.
18. ### I Proof about injectivity

For x to be an element of f(E) it would have to be equal to some f(x) in f(E), right? Just based on the definition. I'm sorry if I'm being slow.
19. ### I Proof about injectivity

The definition of f(E) is {f(x) : x is in E}.
20. ### I Proof about injectivity

Well if the function is injective then we know that x maps to at most one y = f(x). So I don't think this could be true without inectivity.
21. ### I Proof about injectivity

Thank you for the reply! I always forget that to show equality after I have to clearly show that each set is a subset of the other :/ The only next step I can think of is that if f(x) is in f(E), then x is in E, right?
22. ### I Proof about injectivity

I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...
23. ### Would someone mind checking my proof? Intro Real Analysis

LOL! Thanks for checking XD
24. ### Would someone mind checking my proof? Intro Real Analysis

Here is my solution. I used mathjax to type it up in Overleaf. I feel like it makes sense, but I also have a feeling I might have "jumped the gun" with my logic. If it is correct, I would appreciate feedback on how to improve it. Thanks!
25. ### Need help on a proof from Spivak's Calculus

I think I took to many steps in the wrong direction. I just kept reducing. I see how you got what you did, I'm going to reassess.
26. ### Need help on a proof from Spivak's Calculus

I started with \sqrt{ab} \leq \frac{a+b}{2} and just went from there like you said in your original message.
27. ### Need help on a proof from Spivak's Calculus

This is a skill I have been learning very slowly, but surely. It's been especially useful for some set theory proofs I've done recently.
28. ### Need help on a proof from Spivak's Calculus

When I did that, I ended up with the statement 2 \leq a+b , which must be true given what's stated in the problem.
29. ### Today I learned

With great power comes great responsibility!
30. ### Need help on a proof from Spivak's Calculus

Thanks for the hint. It helped a bit.