I've just recently started working through proofs for what's really the first time in my life. Throughout high school, and thus far in college I've never really had to prove things too often, and if I did have to I could just memorize the steps and follow the logic of that specific problem. But now that I'm actually having to deliberately prove (a lot of) things for the first time, I've been feeling like I'm just poking around in the dark, looking for things that'll cancel, factoring, etc... with no real direction. Until I realized something today about proof-solving in general. So I come to you, asking if my logic is correct (for a problem of the form given here).
Say we're supposed to prove this statement:
$a < b$
Where $a$ and $b$ can be composed of $x$'s, $y$'s, $z$'s, $k$'s, $n$'s, or any assortment of variables.
And then say we're given various inequalities, equalities, or other properties (represented by $P$) that delineate the relationships between our variables
$P_1 < P_2$
$P_1 < P_3$ and
If we assume that the initial statement is indeed true, and that we're not just having our jimmies rustled, then what we're really trying to do is to compose some valied third statement "$\varepsilon$" from our given information such that
$a \leq \varepsilon < b$
$a < \varepsilon \leq b$
However, there may not be just one $\varepsilon$ that you can say fits between $a$ and $b$, so it may be necessary to come up with a string of inequalities/equalities such that
I realized this today because I was trying to work through a problem from Spivak's Calculus, and I kept coming up with statements such that
$\varepsilon <a < b$ and it wasn't getting me anywhere closer to proving that $a$ is less than $b$. But by using my various $P$'s to construct an argument that I knew was greater than $a$ but less than $b$, I was finally able to solve it.
And as a final thought, I imagine that in later mathematics there may be properties of numbers that I'm not yet aware of, and these proof-solving ideas may not apply. Or if we're trying to prove that a certain property applies to a argument then this obviously won't work either. But for something like with what we've been presented, does this logic make sense?