You are given $n$ ($n$ is even) integers $a_0,a_1,\ldots,a_{n-1}$ representing temperature measurements, equally spaced around a circle. Since the points are "close", the temperature difference between two points next to each other is at most one. In other words, $|a_i-a_{i+1}|\leq 1$(for $i=0,\ldots,n-2$) and $|a_{n-1}-a_0|\leq1$.
Two points $i,j$ are opposite if $|i-j|\equiv n/2 \mod n$ (i.e. literally opposite on the circle). Define a function $D(i):=a_i-a_j$, where $j$ is the opposite point of $i$ (i.e. $j=i+n/2 \mod n$).
I'm trying to prove that there always exists a point $i$ such that $|D(i)|\leq 1$.
I think proof by contradiction is a viable option, and for that I would assume that $\forall i,|D(i)|\geq 2$. However, I don't really know how to condense my random ideas into a working proof for this problem. Can anyone help me? This is homework by the way.