I'm a bit lost on this example that was worked in class as our professor essentially used the famous "it's easy to show" explanation of the given property,
If $x,y \in \mathbb R$ and $x \leq y$, then $[x]\leq[y]$ where $[x],[y]$ are the greatest integer functions.
I understand this property and it seems very straightforward though I'm not sure how one would go and prove this, thanks.
Notice that for all $z \in \mathbb{Z}$ $$ [z]=max\{m \in \mathbb{Z} ; m \leq z\}. $$ So if $x \leq y$ we have $$ \{m \in \mathbb{Z} ; m \leq x\} \subset \{m \in \mathbb{Z} ; m \leq y\} $$ Then $$ [x]=max\{m \in \mathbb{Z} ; m \leq x\}\leq max \{m \in \mathbb{Z} ; m \leq y\}=[y]. $$