Priving that the degree of an irreducible nondegenerate subvariety of $\mathbb{P^n}$ satisfies $d\ge n-k+1$

Question

I'm reading through Dolgachev's "Classical algebraic geometry" and theorem 8.1.1 states the following

The references are not very precise, do you happen to know where, precisely, I can find the proof in any source?

Answer 1

One source is On Varieties of Minimal Degree by Eisenbud and Harris. This is available here at MSRI's website, for instance. The bibliography lists previous proofs by Bertini (1907), Harris (1981), and Xambo (1981), if you're looking for other sources.