Epsilon-N

Apr 2010
30
0
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.

Thanks
 

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May 2010
274
67
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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