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It suffices to show that you can replace the original sequence $u_k$ v_k$by a new one that is bounded. You should start by figuring out what can go wrong, i.e, how is it possible for the sequence$u_k$v_k$ to be unbounded?

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It suffices to show that you can replace the original sequence $u_k$ by a new one that is bounded.

You should start by figuring out what can go wrong, i.e, how is it possible for the sequence $u_k$ to be unbounded?