Thread: bounded sequence proof

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)

Originally Posted by luckyc1423

A) Suppose that Sn = 0, if (Tn) is a bounded sequence, prove that lim (SnTn) = 0

We need to show,

|s_n*t_n|<e

Note that |t_n|<=M (where M can be chosen to be non-zero).

Then,

|s_n*t_n|<=M*|s_n|<e thus, we need that, |s_n|<e/M

Which is possible for some n>N because lim s_n = 0.
Q.E.D.

Originally Posted by luckyc1423

B) Show by an example that the boundeness of (Tn) is a necessary condition in part

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.

Originally Posted by ThePerfectHacker

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.