Prove that for all real numbers $a$ and $b$, if $0 < a < b$ then $0 < a^2 < b^2$

Question

I'm trying to prove:

For all real numbers a and b, if $0 < a < b$ then $0 < a^2 < b^2$

This is my first class in formal proofs, and I'm not sure where to begin. I'd appreciate any help

Answer 1

Look at your list of axioms regarding $\lt$. You should have one that says something like if $c,d,e \gt 0, d \lt c,$ then $ed \lt ec$ You should also one that says $\lt$ is transitive, that if $c \lt d$ and $d \lt e$ then $c \lt e$. You want to use these by finding something between $a^2$ and $b^2$ that you can relate to both.