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Math Help - Sequences

  1. #1
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    Sequences

    Suppose the lim Sn = 0. If (Tn) is a bounded sequence, prove that lim(SnTn)=0.

    Give a counterexample to show boundedness of (Tn) is necessary.
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  2. #2
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    \begin{gathered}<br />
  \varepsilon  > 0 \hfill \\<br />
  \left( {\exists B > 0} \right)\left( {\forall n} \right)\left[ {\left| {T_n } \right| \leqslant B} \right] \hfill \\<br />
  \frac{\varepsilon }<br />
{B} > 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \left| {S_n } \right| < \frac{\varepsilon }<br />
{B}} \right] \hfill \\\end{gathered}
    \begin{gathered}<br />
  \left| {S_n T_n } \right| \leqslant \left| {S_n } \right|\left| {T_n } \right| < B\frac{\varepsilon }<br />
{B} = \varepsilon  \hfill \\ <br />
\end{gathered}
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