This is from page 12 of Putnam and Beyond.
Problem:
Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that
$\left| m + \sum\limits_{s\in S} s\right| \leq \frac{1}{n+1}$
Solution:
Recall that the fractional part of a real number x is $x − \lfloor{x} \rfloor$ Let us look at the fractional parts of the numbers $x_1, x_1 + x_2, \ldots,x_1 + x_2 +\cdots+x_n$. If any of them is either in the interval $[0, \frac{1}{n+1}]$ or $[\frac{n}{n+1}, 1]$, then we are done. If not, we consider these n numbers as the “pigeons’’ and the $n − 1$ intervals $[\frac{1}{n+1}, \frac{2}{n+1}]$, $[\frac{2}{n+1}, \frac{3}{n+1}],\ldots, [\frac{n-1}{n+1}, \frac{n}{n+1}]$ as the “holes.’’ By the pigeonhole principle, two of these sums, say $x_1 +x_2 +\cdots+x_k$ and $x_1+x_2+\cdots+x_k+m$, belong to the same interval. But then their difference $x_k+1+\cdots+x_k+m$ lies within a distance of $\frac{1}{n+1}$ of an integer, and we are done.
My Question:
I am pretty lost in this question and I can't really see how the solution connects with the problem. I would appreciate it if someone could simplify/break it down for me.