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...
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...
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!
Homework Statement
I'm currently working through Spivak independently and have reached the problems at the end of ch. 1.
The problem is:
Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b
Homework Equations
Spivak's properties P1 - P12
The Attempt at a Solution
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