# Epsilon-N

#### Mathman87

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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#### 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).

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