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Sun 16 Oct, 2005 05:17 pm
Hi, I'm having a hard time proving limits using the convergence definition.
for all (epsilon) > 0 there exists an No such that if n>=No, then |Xn - x| <(epsilon)
I have three problems that I can't figure out....
Prove the limit of ((3+1/n)^2) is 9. I know to take the absolute value of ((3+1/n)^2 - 9) but after that it isn't too clear.
The other two problems are
Let Xn>=0 for each n. Let x be in the real numbers with Xn --> x. Note that x>=0. SHOW that the square root of Xn --> the square root of x. The book says to do 2 cases with x=0 and x>0 and says to rationalize. When I rationalize, I get the absolute value of [(Xn - x) / Sqrt(Xn) + Sqrt(x)], and then am stuck.
The other problem is For all n element of the natural numbers with
Xn-->x and Yn-->y , show x<=y.
Any help would be appreciated, maybe to clarify what you are supposed to do, these seem very different then the examples that we went over in class, I just can't seem to get a grip on them.
For the first problem, don't you just have to show that you can make 6/n + 1/(n^2) arbitrarily small?
For the second question, if x>0, then for every sufficiently large n, Xn > 0 and it holds that
Code:
|sqrt(Xn) - sqrt(x)|
=|Xn - x| / (sqrt(Xn) + sqrt(x))
<= |Xn - x| / sqrt(x)
and you can easily find the proof.
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