The Lonely Runner conjecture states that each runner is lonely at some point in time. Let the speeds of the runners be 
As we know, the distance between 
![t\in \left[\frac{1}{k|a_e-a_i|}+\frac{n_i}{|a_e-a_i|}, \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right)\frac{n_i}{|a_e-a_i|}\right]](https://s0.wp.com/latex.php?latex=t%5Cin+%5Cleft%5B%5Cfrac%7B1%7D%7Bk%7Ca_e-a_i%7C%7D%2B%5Cfrac%7Bn_i%7D%7B%7Ca_e-a_i%7C%7D%2C+%5Cleft+%281-%5Cfrac%7B1%7D%7Bk%7D%5Cright+%29%5Cleft%28%5Cfrac%7B1%7D%7B%7Ca_e-a_i%7C%7D%5Cright%29%5Cfrac%7Bn_i%7D%7B%7Ca_e-a_i%7C%7D%5Cright%5D&bg=FFFFFF&fg=111111&s=0&c=20201002 "t\in \left[\frac{1}{k|a_e-a_i|}+\frac{n_i}{|a_e-a_i|}, \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right)\frac{n_i}{|a_e-a_i|}\right]")
Now we prove the loneliess of
![\bigcap_{i\in A} \left[\frac{1}{k|a_e-a_i|}+\frac{n_i}{|a_e-a_i|}, \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right)\frac{n_i}{|a_e-a_i|}\right]](https://s0.wp.com/latex.php?latex=%5Cbigcap_%7Bi%5Cin+A%7D+%5Cleft%5B%5Cfrac%7B1%7D%7Bk%7Ca_e-a_i%7C%7D%2B%5Cfrac%7Bn_i%7D%7B%7Ca_e-a_i%7C%7D%2C+%5Cleft+%281-%5Cfrac%7B1%7D%7Bk%7D%5Cright+%29%5Cleft%28%5Cfrac%7B1%7D%7B%7Ca_e-a_i%7C%7D%5Cright%29%5Cfrac%7Bn_i%7D%7B%7Ca_e-a_i%7C%7D%5Cright%5D&bg=FFFFFF&fg=111111&s=0&c=20201002 "\bigcap_{i\in A} \left[\frac{1}{k|a_e-a_i|}+\frac{n_i}{|a_e-a_i|}, \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right)\frac{n_i}{|a_e-a_i|}\right]")
The Lonely Runner conjecture states that each runner is lonely at some point in time. Let the speeds of the runners be
As we know, the distance between
Now we prove the loneliess of
