Apr 2010
See Q6 on attached document....

Part (a) is just a definition and is simple enough.

Part (b) i got an answer 10/64\(\displaystyle \epsilon\) using the "big top/small bottom" technique. Was this the right way to do it?

Part (c) i have no idea how to tackle these types of questions so was hoping someone would be able to walk me through the theory for them.



May 2010
Los Angeles, California
I'm not sure your answer for part (b) is right. How did you obtain it? I got \(\displaystyle N=\frac{1}{2\epsilon }\).

Part (c) is easy enough:

(i)\(\displaystyle |a_n-a|<a/2 \leftrightarrow -a/2 < a_n-a <a/2.\) Now add \(\displaystyle a\) to both sides.

(ii) This is just using the difference of two squares:

\(\displaystyle |a_n-a| = |(\sqrt a_n -\sqrt a)(\sqrt a_n+\sqrt a)|\)

From (i) we have that \(\displaystyle \sqrt a_n > \sqrt{a/2}\) and so \(\displaystyle \sqrt a_n +\sqrt a> \sqrt{a/2} + \sqrt a . \) and the result follows (I think \(\displaystyle \le\) is a typo).

Part (c) is needed for part (d).
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