There exist arbitrarily large prime gaps

Statement

For every positive integer $m$, there exists a sequence of $m$ consecutive integers all of which are positive. Thus, there exists a prime gap between consecutive primes that is greater than $m$.

Proof

Let $n=m+1$. Consider the integers $n!+2,n!+3,\dots ,n!+n$. For each $2\le i\le n$, $n!+i$ is divisible by and strictly larger than $i$, hence is composite. Further, the sequence has length $n-1=m$.