I am taking a first class in real analysis and am stumped already. The text leaves out the steps to the following relationship and I can not connect the dots. Any explanations would be appreciated.
(m,n) $\simeq$ (p,q) and (p,q) $\simeq$ (r,s) $\Rightarrow$ (m,n) $\simeq$ (r,s)
I have no problem showing (m,n) $\simeq$ (p,q) $\Rightarrow$ (p,q) $\simeq$ (m,n)
$\ (m,n) \simeq (p,q) \Rightarrow m+q=n+p \Rightarrow q+m=p+n \Rightarrow p+n=q+m \Rightarrow (p,q) \simeq (m,n)$
but I can not make the next step.
Thanks for looking.
Suppose that $$(m,n)\simeq(p,q)\quad\hbox{and}\quad (p,q)\simeq(r,s)\ .$$ According to your definition this means that $$m+q=n+p\quad\hbox{and}\quad p+s=q+r\ .$$ Adding these equations, $$m+q+p+s=n+p+q+r\ ;$$ cancelling $q$ and $p$ gives $$m+s=n+r\ .$$ By definition this means $$(m,n)\simeq(r,s)\ .$$