Given 10 real numbers $x_1,x_2, \ldots ,x_{10}$. For each $i \in \{1,2, \ldots ,10\}=S$ define
$(1) \;\; r_i = x_i - [x_i] \in [0,1)$.
Then for two different integers $i,j \in S$
$(2) \;\; x_i - x_j$ is almost an integer iff $|r_i - r_j| \in [0,0.1] \cup [0.9,1)$.
Let $y_1, y_2, \ldots ,y_{10}$ be the permutation of $r_1,r_2, \ldots ,r_{10}$ s.t.
$(3) \;\; y_1 \leq y_2 \leq \ldots \leq y_9 \leq y_{10}$.
Next assume $y_{i+1} - y_i$ is not an almost integer for every $i \in \{1,2, \ldots ,8,9\}$. Hence by equivalence (2) and condition (3)
$\sum_{i=1}^9 0.1 < \sum_{i=1}^9 |y_{i+1} - y_i| = \sum_{i=1}^9 (y_{i+1} - y_i)$,
i.e.
$y_{10} - y_1 > 0.9$,
which according to equivalence (2) means the difference $x_i - x_j$, where $r_i=y_{10}$ and $r_j=y_1$, is almost an integer. q.e.d