Suppose {an }n≥1 , {bn }n≥1 are Cauchy sequences of real numbers. Carfeully prove that {anbn }n≥1 is Cauchy.

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- Sep 20th 2010, 05:21 PMcalculuskid1Cauchy sequences
Suppose {an }n≥1 , {bn }n≥1 are Cauchy sequences of real numbers. Carfeully prove that {anbn }n≥1 is Cauchy.

No idea where to start or what to do - Sep 21st 2010, 03:34 AMHallsofIvy
$\displaystyle |a_nb_n- a_mb_m|= |a_nb_n- a_mb_n+ a_mb_n- a_mbm|\le |a_nb_n- a_mb_n|+ |a_mb_n- a_mb_m|$$\displaystyle = |b_n||a_n- a_m|+ |a_m||b_n- b_m|$.