# Epsilon-N

• May 13th 2010, 06:55 PM
Mathman87
Epsilon-N
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
• May 13th 2010, 08:40 PM
ojones
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).