I am reading the proof of Turan's theorem on $ex(n,K_r)$ in Diestel's Graph Theory book as follows.
Question: I am confused about the induction. The normal induction is that we first verify that $n=1$ is true. Assume that $n=k$ is true and verity $n=k+1$ is also true. However, in this proof we have known that $n\le r-1$ is true. What is the assumption here? Do we assume that $n=r$ is true? I am confused about the sentence
By the induction hypothesis, $G-K^{r-1}$ has at most $t_{r-1}(n-r+1)$ edges?
