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Real analysis: Limit of a product of sequences

  1. Nov 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Let (u[itex]_{n}[/itex])[itex]_{n}[/itex] be a real sequence such that lim u[itex]_{n}[/itex] = 0 as x→∞ and let (v[itex]_{n}[/itex])[itex]_{n}[/itex] be a bounded sequence. Show that lim (u[itex]_{n}[/itex])[itex]_{n}[/itex](v[itex]_{n}[/itex])[itex]_{n}[/itex] = 0 as x→∞

    2. Relevant equations

    3. The attempt at a solution

    Since (v[itex]_{n}[/itex])[itex]_{n}[/itex] is bounded then it has a least upper bound and greatest lower bound. Then we have g.l.b< lim (v[itex]_{n}[/itex])[itex]_{n}[/itex] <l.u.b
    I don't really know how to take it from here. Does the existence of the limit of (u[itex]_{n}[/itex])[itex]_{n}[/itex] mean it is bounded?
    Last edited: Nov 15, 2011
  2. jcsd
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