|This is an AoPSWiki Word of the Week for May 4-11|
Suppose is between and for all in the neighborhood of . If and approach some common limit L as approaches , then .
If is between and for all in the neighborhood of , then either or for all in the neighborhood of . Since the second case is basically the first case, we just need to prove the first case.
For all , we must prove that there is some for which .
Now since, , there must exist such that,
Now let . If , then
So . Now by the definition of a limit, we get .