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**Plato** You are to prove that given any $\displaystyle \varepsilon >0$ it is possible to insure that $\displaystyle

\left| {a_n b_n - 0} \right| < \varepsilon $.

So from the fact that $\displaystyle \left( {a_n } \right) \to 0$ you use the definition of sequence convergence.

But in place of just $\displaystyle \varepsilon >0$ you are free to use $\displaystyle \frac{\varepsilon}{M} >0$.

$\displaystyle \left| {a_n b_n - 0} \right| = \left| {a_n b_n } \right| = \left| {a_n } \right|\left| {b_n } \right| \leqslant \left| {a_n } \right|M < \frac{\varepsilon }{M}M = \varepsilon $