Suppose that (sn) and (tn) are convergent sequences with
lim sn =s and lim tn =t. Then lim (sntn)= st. Prove this without using the identity sntn-st= (sn-s)(tn-t) +s(tn-t)+t(sn-s).
This is the complete solution.
$\displaystyle \left| {s_n t_n - ST} \right| \leqslant \left| {s_n t_n - St_n } \right| + \left| {St_n - ST} \right| \leqslant \left| {t_n } \right|\left| {s_n - S} \right| + \left| S \right|\left| {t_n - T} \right|$
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