how to prove that any countable limit ordinal is really a limit?

If ordinal number , then we call a successor ordinal. An ordinal is a limit ordinal if it is not a successor. My question is: how to prove that for any countable limit ordinal , there is a sequence in converging to ? Here we define a sequence in to be a function on the set of all natural numbers and having values in , and a sequence of ordinals converging to ordinal if for any , there is a natural number such that for all .

For example, sequence , .... But these are specific sequences, I need a proof in general situation.

Thanks!