Prove the following statement:
For every $\lambda$>1, there exists a number a∈N and b∈[0,1) such that $\lambda$=a+b.
I first defined a = sup{n ∈ N | n ≤ x}, so m is the integer part of x or the floor of ⌊x⌋. How would i rigorously prove the floor of x is the sup? Then I need to prove x-⌊x⌋=x-a<1, is this trivial or do I need to prove it too?
In your other question you asked how to show
a real r >= 1 is in [n,n+1) for some n in N.
Use that result to conclude n is the integer part of r
and r - n the fractional part.