A) Suppose that Sn = 0, if (Tn) is a bounded sequence, prove that lim (SnTn) = 0
B) Show by an example that the boundeness of (Tn) is a necessary condition in part (A)
Theorem A convergent sequence is bounded.
We have shown that,
lim (s_n*t_n)=0
Thus,
|s_n*t_n|<= M
By the previous theorem.
BUT! {s_n} is also bounded by some non-zero konstant K because it is convergent.
Thus,
|s_n*t_n|<=K|t_n|<=M
Thus,
|t_n|<=M/K
Thus,
{t_n} must be a bounded sequence.
Q.E.D.
After I posted this yesterday I realized I made a mistake.
I will not tell you were, you shall need to find it yourself.
But what you said is false.
Consider,
s_n=0.
And
t_n=n
Note,
lim (s_nt_n)=0.
But {t_n} is not bounded!
I will state the following as an excercise (a little hard to show). But try it.
Theorem Let lim (s_n) not = 0 and s_n not =0. Let t_n be any sequence. If {s_n*t_n} is a convergent sequence, then {t_n} must be bounded.