I'm reading Courant's Differential and Integral Calculus, there is this session of exercises:
My trouble here is that previously he talks about the rational numbers, why it is needed for them to be extended, he talks a little bit about real numbers and their representation as infinite decimals, expression of numbers in scales other than $10$, inequalities and the Schwarz's Inequality. My question is: Are we supposed to prove all these using only what was given before in the book? We have two classes of exercises:
Exercises in which it is obvious that you need what was given previously: It is quite obvious that to solve - for example - $6,8,etc$ you can use what was given in the sections about inequalities, Schwarz' inequality, etc.
Exercises in which it is not obvious that you need what was given previously: For example, to solve $1.a$, I used the common proof with factorization of integers, $1d$ needs the test of rational roots, which was not given previously unless it was under some disguise, $2$ might need something else but I have been thinking if the Schwarz' inequality could be used to prove it, I have been doing the same (Schwarz) for the other examples without success.
This is important for me because I'm not sure what tools I am allowed to use and I know that there could be some out-of-the-box technique in which it's much easier to prove things, for example: For the examples in $5c$, I could use maxima/minima from single-variable calculus, for $5a,c$ I could use maxima/minima from multivariable calculus, but that would take all the fun from it.