Ceiling of Half of n+m plus Ceiling of Half of n-m+1

Theorem

Let $n, m \in \Z$ be integers.

$\ceiling {\dfrac {n + m} 2} + \ceiling {\dfrac {n - m + 1} 2} = n + 1$

where $\ceiling x$ denotes the ceiling of $x$.

Either $n + m$ or $n - m + 1$ is even.

Thus either $\dfrac {n + m} 2$ or $\dfrac {n - m + 1} 2$ is an integer.

So:

$\ds \ceiling {\dfrac {n + m} 2} + \ceiling {\dfrac {n - m + 1} 2}$

$=$

$\ds \ceiling {\dfrac {n + m} 2 + \dfrac {n - m + 1} 2}$

$\ds $

$=$

$\ds \ceiling {\dfrac {n + m + n - m + 1} 2}$

$\ds $

$=$

$\ds \ceiling {n + \dfrac 1 2}$

$\ds $

$=$

$\ds n + 1$

$\blacksquare$