You may want to wait for confirmation on this...it may not be rigorous enough....also how late it is may cause concern for incorrectness.
I think you alternatively to TPH's response may be able to say that since is a bounded sequence that the set of subsequential limits of is a bounded subset of the reals. This fact lets us apply all the basic "rules and theorems" of subsets of the reals to . So now using TPH's first remark and noting that and we can draw our conclusion.