My textbook (Bostock and Chandler - Pure Mathematics 1) says something that really surprises me.

The bolded part is what shocked me, it seems like such a trivial statement and intuitively true. My book was first published in 1978, so perhaps it is out of date.

It goes on to say;

Which seems contradictory... So, has or has not this idea been proven true?

I mean that it seems trivial, so I was surprised that it had not been proven true. By contradictory, I mean, the book says the idea is not generally proven but goes on to say that it is true for all values of k.

It is not explicitly stated in my textbook what is meant by k, but all related questions deal with positive numbers, fractions and integers.

Perhaps you're taking the context of the bolded statement to be total human mathematical development, rather than the mathematical development up to that point in the text?

I agree with Integrand. It sounds like the textbook authors want to make it clear that they are not providing a proof. They are distinguishing a conjecture making moment. If the text takes an investigation approach, then it probably encourages readers to do similar activities to develop conjectures and then better proofs.

The part that you say is contradictory is what I would call Proof by Authority. These are moments in textbooks where the author just asks the reader to accept the math without other justifications. This is often necessary because a proof requires advanced mathematics or may take too long. There's a lot of this in algebra texts: fractional exponents, calculating determinants, formulae of SA and volume of spheres. Typically the reader is just given these rules.