Show that the statement sn converges to L is false if and on

Show that the statement \"{s_n} converges to L\" is false if and only if there is a positive number c so that the inequality |s_n - L| > c holds for infinitely many values of n.

Solution

Assume, to the contrary, that \"{sn} converges to L\" is false and for every c>0c>0, the inequality

Let c>0 be given. Since |snL|>c holds for only finitely many n, let N be the largest integer for which |sL|>c holds. Then n>N implies

|snL|c..

we deduce that {sn} converges to L, which would contradict our assumptions. But in order to prove convergence,

we need to show that for every c>0there is an integer N such that nN implies |s |snL|<c. From what we have been able to deduce, we can show that for every c>0 there is an integer N such that nN implies |snL|c, but we cannot see how to make this inequality strict.

we thought about breaking it into three cases: