[PLAIN]http://img839.imageshack.us/img839/2670/hardlimitproblem.jpg [Broken]

I am having trouble showing this. I figured I would divide both sides by a, and show that

lim

I can explain in words why I believe this to be true; as n goes to infinity, there are infinitely many terms of a

Help please :)

Edit: I think I have managed to bound it below...I'm not sure its legitimate though..

Let a

(1+e/a)*(b

If this is right, then I still need to bound this above by 1, which I can't figure out how to do, since taking the largest of a

I don't think what I did works though, because I think my argument gets messed up if some of the a

I am having trouble showing this. I figured I would divide both sides by a, and show that

lim

_{n-> infinity}[(a_{1}/a)*b_{1}+....+(a_{n-1}/a)*b_{n-1}+(a_{n}/a)*b_{n}]/(b_{1}+...+b_{n}) = 1.I can explain in words why I believe this to be true; as n goes to infinity, there are infinitely many terms of a

_{i}/a = 1, because as i gets really big it approaches a. Since there are infinitely many of these, and the denominator is infinite, the a_{i}/a terms that don't approach one are negligible. The problem is I can't figure out how to translate this into math. I was thinking of somehow doing an epsilon argument, but I'm not sure how I would do that.Help please :)

Edit: I think I have managed to bound it below...I'm not sure its legitimate though..

Let a

_{n}-a = e such that |a_{n}-a| is the smallest of the |a_{i}-a| values. Then with some algebra, we find a_{n}/a = 1+e/a...replacing all of the a_{i}/a's in my equation with 1+e/a, we get(1+e/a)*(b

_{1}+..+b_{n})/(b_{1}+..+b_{n)}) = 1+e/a, taking the limit as n goes to infinity, this equals 1, because a_{n}converges to a. Since 1+e/a is the smallest of the a_{i}/a values, 1 < lim_{n-> infinity}[(a_{1}/a)*b_{1}+....+(a_{n-1}/a)*b_{n-1}+(a_{n}/a)*b_{n}]/(b_{1}+...+b_{n})...If this is right, then I still need to bound this above by 1, which I can't figure out how to do, since taking the largest of a

_{i}/a doesn't work.I don't think what I did works though, because I think my argument gets messed up if some of the a

_{i}/a are negative.
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