I just started real analysis. I don't have a background in proofs or logic, simply calculus. So I'm trying to learn more about proofs--so forgive the basic question, please.
How do you go about proving this theorem: If $a$ and $b$ are any numbers, then there is one and only one number $x$ such that $a + x=b$. This number is given by $x=b+(-a)$.
First part of the proof I understand--it's simple. We simply use some axioms and do the following:
$a+b-a=b$, which of course is true. This is just true from plugging in the $b+(-a)$ for $x$.
But how about the uniqueness issue? In my text, it says:
$$(a+x)+(-a)=b+(-a) x=b+(-a)$$
How does this prove uniqueness? Is this equation essentially saying that the ONLY possible value of $x$ is expressed by $b+(-a)$?
How would I have known to set this up as part of the proof if I was asked to prove something is unique?
Sorry for the basic question--have to start somewhere.