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!