Let $p_n$ denotes the $n$-$th$ prime number ,when we enumerate the prime numbers in the increasing order. For example ,$p_1=2,p_2=3,p_3=5$,and so on. Let $S$={${s_n=p_{n+1}-p_n \mid n\in \mathbb{N} ,n\geq 1}$}
Then $\underline{\lim}_{n \rightarrow\infty}\; s_n \geq 2 $
I know that inf of set $S$ is $1$ ,but how the $\underline{\lim}_{n \rightarrow}\infty\; s_n \geq 2 $ is possible, as I know some consecutive prime that has difference of $2$, but not confirm that there are that much consecutive primes with difference $2$, so that it becomes limit point.
Please help.
Thank you.