I want to know if my attempt is anywhere near being correct...

I'm supposed to prove that floor(n + m) = n + floor(m) where n is an integer and m is a real number. I'll write this as [n + m] = n + [m].

[n + m] can be rewritten as floor(( n + [m] ) + ( m - [m] )).

By definition, [m] is the largest integer not greater than m. Therefore the absolute value of ( m - [m] ) is less than 1.

n is an integer and [m] is an integer, so ( n + [m] ) is also an integer because the set of integers is closed under addition.

So floor(( n + [m] ) + ( m - [m] ))or floor(n + m) is equal to the integer ( n + [m] ), because this is the largest integer not greater than (n + m).

Therefore [n + m] = n + [m].

Thank you!