**I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.**

## This Week

In lectures, we started chapter 3 and are in section 3.2. We also looked at some simpler proofs of Theorems 1.3.2 and 3.1.4. These are reproduced below.

In tutorials we did p.36 Q 6, p.42 Q.2, 4 and p.44 Q.1 (a)(e), 2(a).

## Sample Test

Hopefully by Tuesday morning.

## Additional Notes

Here we give an alternative proof of Theorem 3.1.4. It is very likely that you will be asked to prove some of Theorem 3.1.4 in the exam and you are free to choose which proof you prefer. You may even prove parts (ii) and (iii) of Theorem 3.1.4 using the proof in the notes and then use this proof for part (iv). In class I also gave an easier proof of Theorem 1.3.2 although this won’t be examinable.

### Theorem 3.1.4

**For all positive and , and any rational number , ***we have*

*,**,**,**.*

*Proof *: The proof of 1. is trivial.

2. Consider the functions

and .

Note that and . Therefore and have the same slope so by Theorem 1.3.2 they only differ by a constant, say

for all .

Now let :

.

for all .

Now let ;

.

3. Now by 2. we have that

. (*)

However

.

But by 1. Hence

.

Putting this back into (*):

.

4. Consider the functions

and .

Note that and . Hence these functions have the same slope and thus differ only by a constant, say :

for all .

Let ;

.

Thence for all and particular for any :

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