Limit Theorems, Sequences

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March 26th 2009, 12:38 PM
bearej50

Limit Theorems, Sequences

Suppose that (sn) and (tn) are convergent sequences with lim sn = s and lim tn = t. Then lim (sntn) = st.

Write a proof to this theorem that does not use the theorem that states that "every convergent sequence is bounded." I think that this can be done by using the identity sntn - st = (sn - s)(tn - t) + s(tn - t) + t(sn - s). I am lost from here.
March 26th 2009, 01:02 PM
Plato

If $\varepsilon > 0$ then choose $\delta = \min \left\{ {\sqrt {\frac{\varepsilon }{3}} ,\frac{\varepsilon }<br /> {{3\left( {\left| s \right| + 1} \right)}},\frac{\varepsilon }<br /> {{3\left( {\left| t \right| + 1} \right)}}} \right\}$ .

Make $\left| {s_n - s} \right| < \delta \;\& \,\left| {t_n - t} \right| < \delta$ .

Then use the identity.
March 27th 2009, 06:33 AM
bearej50

I'm still lost...
March 27th 2009, 04:09 PM
xalk

Quote:

Originally Posted by bearej50

I'm still lost...

The identities that you must use are:

1) $s_{n}t_{n}-st = s_{n}t_{n}-st_{n}+st_{n}-st = t_{n}(s_{n}-s)+s(t_{n}-t)$

OR

2) $s_{n}t_{n}-ts_{n}+ts_{n}-st = s_{n}(t_{n}-t)+t(s_{n}-s)$ .

In using for example the (1) identity we can work as follows:

we want to find out if there exists a No ,k, belonging to the natural Nos,such that for any given positive No ε>0

$|s_{n}t_{n}-st| = |s_{n}t_{n}-st_{n}+st_{n}-st| = |t_{n}(s_{n}-s)+s(t_{n}-t)|<\epsilon$ ,whenever , $n\geq k$ .

WHERE n is any natural No.

But since $|t_{n}(s_{n}-s)+s(t_{n}-t)|\leq |t_{n}||s_{n}-s| + |s||t_{n}-t|\leq |t_{n}||s_{n}-s| + (|s|+1)|t_{n}-t|$ ,if we can find a ,k, such that for :

$n\geq k$ ,then $|t_{n}||s_{n}-s|<\frac{\epsilon}{2}$ ,and

$(|s|+1)|t_{n}-t|<\frac{\epsilon}{2}$ ,we will have proved the desired result.

Note instead of using |s| we use ||s|+1| in the above inequalities is to avoid the case where s=0.

HINT: In trying to find this ,k, it must be the max{ $n_{1},n_{2},n_{3}$ } i,e the maximum of three other natural Nos
March 30th 2009, 04:21 PM
xalk

bearej50.

Since there is no respond from you ,i will assume that you still do not know how to do the limit,so i will prove it.

So we want to prove $\lim_{n\rightarrow\infty}{s_{n}t_{n}} = st$

Let ε>0.

Thus $\frac{\epsilon}{2(|t|+\epsilon)}>0$ .................................................. ................................1

and $\frac{\epsilon}{2(|s| +1)}>0$ .................................................. ........................................2

Since now $\lim_{n\rightarrow\infty}{s_{n}} = s$ and $\lim_{n\rightarrow\infty}{t_{n}} = t$ ,given any +ve No and thus even the +ve Nos (1),(2),ε,
we can find Natural Nos $n_{1},n_{2},n_{3}$ and such that:

If $n\geq n_{1}$ ,then $|t_{n}|<|t|+\epsilon$ .................................................. ...........................................3

If $n\geq n_{2}$ ,then $|s_{n}-s|<\frac{\epsilon}{2(|t|+\epsilon)}$ .................................................. .........................................4

If $n\geq n_{3}$ ,then $|t_{n}-t|<\frac{\epsilon}{2(|s|+1)}$ .................................................. ...........................................5

Choose k = max{ $n_{1},n_{2},n_{3}$ }

Hence ,

If $n\geq k$ ,then $n\geq n_{1},n\geq n_{2},n\geq n_{3}$ ,

HENCE

$|t_{n}|<|t| +\epsilon$ .................................................. ........................................6

$|s_{n}-s|<\frac{\epsilon}{2(|t|+\epsilon)}$ .................................................. ........................................7

$|t_{n}-t|<\frac{\epsilon}{2(|s|+1)}$ .................................................. .........................................8

Now multiply (6) by (7) and we get:

$|t_{n}||s_{n}-s|< (|t|+\epsilon)\frac{\epsilon}{2(|t|+\epsilon)} = \frac{\epsilon}{2}$ .................................................. ...........................................9

And by multiplying (8) by (|s|+1) we get:

$(|s|+1)|t_{n}-t|< (|s|+1)\frac{\epsilon}{2(|s|+1)} =\frac{\epsilon}{2}$ .................................................. ..........................................10

Add (9) and (10) we get:

$|t_{n}||s_{n}-s|+(|s|+1)|t_{n}-t|< \frac{\epsilon}{2} +\frac{\epsilon}{2} = \epsilon$ .

Hence $|t_{n}s_{n}-ts|<\epsilon$ .

Therefor we have proved ,given any +ve No ,ε Τhere exist a natural No k such that :

If $n\geq k$ then $|t_{n}s_{n}-ts|< \epsilon$ ,for all n
March 30th 2009, 05:05 PM
Plato

Re the last post.
Did you note that the OP said not to use the bounded theorem ?
Your proof uses it.
March 30th 2009, 05:30 PM
xalk

Nowhere my post uses the the bounded theorem

Read my post again