### My first attempt at solving the Lonely Runner Conjecture

Let us suppose there are $k$ runners running at speeds $0 around a field of circumference $1$. Take any runner from the $k$ runners- say $r_i$, who runs with speed $a_i$. Say we pair him up with another runner $r_j$ who runs with speed $a_j$. Then for time $0\leq t\leq\frac{1}{2|a_i-a_j|}$, the distance between them is $|a_i-a_j|t$, and for $\frac{1}{2|a_i-a_j|}\leq t\leq \frac{1}{|a_i-a_j|}$, the distance between them is $1-|a_i-a_j|t$.

The Lonely Runner Conjecture can be stated in the following way: there exists a time $t$ such that $\min\left[\min\{|a_i-a_1|t_1,1-|a_i-a_1|t_1\},\min\{|a_i-a_2|t_2,1-|a_i-a_2|t_2\},\dots,\min\{|a_i-a_k|t_k,1-|a_i-a_k|t_k\}\right]\geq\frac{1}{k}$.

Here $t_1,t_2,t_3,\dots,t_k$ are the smallest positive values found after successively determining $t_x-n\frac{1}{|a_i-a_x|}$, where $x\in\{1,2,3,\dots k\}\setminus\{i\}$, and $n\in\Bbb{Z}^+$