You can't prove that; for instance for there is no such sequence. (And that makes sense: if there is such sequence, for more convenience assume it strictly increases. Think of the "distance" (sorry for my approximative english, by "distance" between and I mean ) between two successive ordinals in that sequence: if at a moment, it exceeds , then the limit will be greater than . But if all "distances" are at most countable, since a countable reunion of countable sets is countable, then you cannot reach ).
You can find answers with cofinality (Cofinality - Wikipedia, the free encyclopedia )