I understand the concept of what ceilings and floors when using numbers; however, when it comes to proofs I am so lost, I was wondering if anyone could shed some light on some useful links that could help me do proofs with ceilings and floors.
I understand the general definition; however I don't know how to take it from there.
for example: prove $ n =⌊n/3⌋+ ⌈n/3⌉ +⌈n-1/3⌉ $
I know that by definition that x <= ⌊n/3⌋
Mathlove's hint is the solution.
But I'm not Mathlove so I did it the hard way. But this illustrates how I thought about it and came to a conclusion.
Well, let $\frac n 3 = m + a$ where m is an integer and $0<= a < 1$.
As n is an integer, $a = 0$ and $n = 3m$ or $a = 1/3$ and $n = 3m + 1$ or $a = 2/3$ and $n = 3m + 2$.
Then $⌊n/3⌋ = m$.
$⌈n/3⌉ = m + 1$ if a > 0 and $⌈n/3⌉ = m$ if a = 0.
$⌈(n−1)/3⌉ = ⌈n/3 −1/3⌉ =$... well $(n-1)/3 = m + a - 1/3$. So $⌈(n−1)/3⌉ = n$ if a = 0 or a = 1/3. $⌈(n−1)/3⌉ = n + 1$ if a = 2/3
So divide it into cases.
Case 1: $a = 0$. ⌊n/3⌋+⌈n/3⌉+⌈(n−1)/3⌉ = m + m + m = 3m = 3m = n.
Case 2: $a = 1/3$. ⌊n/3⌋+⌈n/3⌉+⌈(n−1)/3⌉ = m + (m + 1) + m = 3m + 1 = n.
Case 3: $a = 2/3$ ⌊n/3⌋+⌈n/3⌉+⌈(n−1)/3⌉ = m + (m + 1) + (m + 1) = 3m + 2 = n.