As stated in an earlier section, here is the required proof. I’ve often seen this ‘left as an exercise for the reader’. Such exercises are fine once you understand the material but not as part of the explanation. One danger is that those who are less fluent with mathematical techniques are left wondering where all this stuff comes from…
About our tutor
Patrick Andrews has a PhD in Mechanical Engineering from HeriotWatt University and an MSc in Neuroscience from the University of Cambridge.
A former Chartered Engineer and Chartered Mathematician, Patrick has worked in management consultancies and hightech startups, and in academia as a postdoc in Applied Mathematics at Cambridge and a maths lecturer at De Montfort University.
Patrick is the father of three delightful (but not uniformly mathematical!) daughters.

Many thanks for all your support. [My daughter] has benefitted enormously.
S5 mum
Thanks for all your help. [My daughter] thinks it’s great.
S5 mum
Thank you very much for getting me through Maths I got my A at the end of the day.
S4 pupil
[Our son] did very well and he is delighted!
S5 mum
Well done – we are all delighted!!
S5 dad
Your methods achieved a step change in [my son’s] academic prowess.
S4 dad
Thank you for all your help with [our daughter]’s maths. An A  she is so chuffed and so am I !!!!
S4 dad
Following all your hard work with my younger son, he was chuffed with his B award in Maths at National 5.
S4 dad
My son was failing his Standard Grade maths. Three months with you and he’s first in his class.
S4 mum
I am very happy to sing your praises! Thanks again.
P4 mum
Higher Maths tutoring was quite simply well structured and professional throughout. Would highly recommend.
S5 dad
Having struggled hugely with Maths …[our daughter] managed to attain a “B” grade in Int 2 Maths. We (and she!) are delighted! Many thanks for your input that I think was instrumental in her getting on so well.
S5 dad
My son is finding the tuition very rewarding… Thanks to you, maths is now on the up.
S4 dad
I got a B for Maths and I'm very pleased!
Thank you very much for boosting my confidence in Maths.
S5 pupil
Thank you for all of your help and support that made passing maths possible for my daughter!
S4 mum