I seem to have unwittingly made the very first question of the homework - Q1(a) of HW5 - one of the hardest questions in the set. First of all, it presumes that one knows how to exponentiate a real number x to an integer n, x^n, but actually in Week 2 notes I only defined how to exponentiate a rational number to an integer n.
However, the definition for real bases x is just the same. Furthermore, since the real numbers obey all the laws of algebra and order that the rationals do, the laws of exponentiation (Propositions 3 and 4 of Week 2 notes) also extend to the real numbers. I've adjusted the Week 2 notes to reflect this. But for the purposes of this homework, you may assume that all the usual laws of exponentiation etc. that you are used to from high-school algebra, are true. (However, you are not allowed to use logarithms or "take n^th roots", since the former has not been defined yet and the latter is circular).
Secondly, I suggested you follow the proof of Proposition 28 from
Week 2 notes in the hint. A direct mimicking of this proof will lead into difficulties because you will find yourself needing the binomial theorem (which can be used, but this requires a lot of machinery, e.g. you need to develop factorials). A slight variant however of the argument is simpler to use. First observe that
lim_{m to infinity} (x + 1/m) = x
and
lim_{m to infinity} (x - 1/m) = x
and hence by induction and limit laws
lim_{m to infinity} (x + 1/m)^n = x^n
and
lim_{m to infinity} (x - 1/m)^n = x^n.
Thus, if x^n < y, then there exists m such that (x + 1/m)^n < y,
and if x^n > y, then there exists m such that (x - 1/m)^n > y.
This turns out to be enough to prove 1(a).
Once you have 1(a), I strongly recommend that you use 1(a) repeatedly (together with the laws of order) to prove 1(b-g).
Terry