I had this question at the start of my number theory class so I think it is supposed to be an easy one but I did not receive a solution. Here is the problem:
Let $n\in \mathbb{N}$ be arbitary. Prove that there exists a number $k\in\mathbb{N}$, such that $k + 1, k + 2, k+3, . . ., k+n\;$ are not primes.
I read somewhere that the gaps between the consecutive primes do not tend towards infinity so is this not a wrong statement? This is what confused me. Can someone shed some light on this?
Let $g_n$ be the sequence of gaps between consecutive primes. The problem posed is to show that $\limsup_{n\to\infty}g_n=\infty$. The Twin Prime conjecture, on the other hand, says that $\liminf_{n\to\infty}g_n=2$. The 2013 breakthrough theorem of Yitang Zhang says that $\liminf_{n\to\infty}g_n\le70{,}000{,}000$ (and follow-up work by others has reduced the upper bound to $246$). Conjecturally, $g_n$ takes every even value infinitely often, but Zhang's theorem alone is enough to show that $g_n$ does not tend to infinity, even though its limsup does.
Added later: Just to be clear, I am answering what I take to be the OP's main question, which is why the assigned problem does not conflict with their having read that the sequence of gaps does not tend to infinity. As the OP surmised, the assigned problem is a relatively easy one (and as such would be a duplicate).