I'm currently reading Lonely runner conjecture which has been proved upto $k=7$. I could understand the cases with $k=1,2$ i.e, one and two runners repsectively, but I am stuck at $k=3$. Can anyone provide me any idea of the proof? (The wikipedia article states it as trivial case, but I can't get it.) Or, can you please provide some reference?
PS: It will be helpful if you can provide ideas/references without relating to View-obstruction problem or Diophantine equation, as I have not grasped how exactly those connects to our conjecture.
We need to show that a stationary runner is lonely at some point. Call the other runners A and B, with A faster than B. Look at the first time B is $1/3$ away from the starting point. If A is at most twice as fast as B, this works. If A is more than twice as fast as B, look at the interval in which B moves from $1/3$ to $2/3$. In this interval A travels more than $2/3$, so at some point he will also be more than $1/3$ from the start.