# Sequences

• November 17th 2008, 08:11 AM
GoldendoodleMom
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.
• November 17th 2008, 08:32 AM
Plato
$\begin{gathered}
\varepsilon > 0 \hfill \\
\left( {\exists B > 0} \right)\left( {\forall n} \right)\left[ {\left| {T_n } \right| \leqslant B} \right] \hfill \\
\frac{\varepsilon }
{B} > 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \quad \left| {S_n } \right| < \frac{\varepsilon }
{B}} \right] \hfill \\\end{gathered}$

$\begin{gathered}
\left| {S_n T_n } \right| \leqslant \left| {S_n } \right|\left| {T_n } \right| < B\frac{\varepsilon }
{B} = \varepsilon \hfill \\
\end{gathered}$