If then choose .
Make .
Then use the identity.
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.
The identities that you must use are:
1)
OR
2) .
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
,whenever , .
WHERE n is any natural No.
But since ,if we can find a ,k, such that for :
,then ,and
,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{ } i,e the maximum of three other natural Nos
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
Let ε>0.
Thus .................................................. ................................1
and .................................................. ........................................2
Since now and ,given any +ve No and thus even the +ve Nos (1),(2),ε,
we can find Natural Nos and such that:
If ,then .................................................. ...........................................3
If ,then .................................................. .........................................4
If ,then .................................................. ...........................................5
Choose k = max{ }
Hence ,
If ,then ,
HENCE
.................................................. ........................................6
.................................................. ........................................7
.................................................. .........................................8
Now multiply (6) by (7) and we get:
.................................................. ...........................................9
And by multiplying (8) by (|s|+1) we get:
.................................................. ..........................................10
Add (9) and (10) we get:
.
Hence .
Therefor we have proved ,given any +ve No ,ε Τhere exist a natural No k such that :
If then ,for all n