About the gap of consecutive prime numbers.

Question

I recently read James Maynard's paper "Small gaps between primes". In his paper, he used the result $$p_{\pi (k)+k}-p_{\pi (k)+1}\ll k\log k.$$ Here $p_n$ denotes n-th prime number and $\pi (n)$ denotes the number of primes up to $n$. If you know the proof or the reference about this result, please tell me.