Hint $\ $ A variant of Euclid's classical proof is that one may construct an infinite sequence of primes from any infinite sequence of coprimes, e.g. give an increasing sequence of naturals $\rm\:f_n > 1\:$ that are pairwise coprime, i.e. $\rm\:(f_i,f_j) = 1\:$ for $\rm\:i\ne j,\:$ then choosing $\rm\:p_i\:$ to be a prime factor of $\rm\:f_i\:$ yields an infinite sequence of primes, since the $\rm\:p_i\:$ are distinct: $\rm\:p_i\ne p_j,\:$ being factors of coprimes $\rm\:f_i,\, f_j\:$.
Therefore $\rm\:f_n \ge \:$ the $\rm n$'th prime, since there are at least $\rm\:n\:$ primes smaller than it, viz. the primes $\rm\:p_1,\ldots, p_n,\:$ where $\rm\:p_k|\:f_{\,k}\:\Rightarrow\:p_k\le f_{\,k} \le f_n\:$ by $\rm\:k\le n,\:$ since $\rm\:f_n\:$ is increasing.
To complete the proof of your problem, you need only show there are two more such primes when $\rm\:f_n = F_n,\:$ which the hint reveals, viz. the prime $2$ and the pair of primes from $\rm\:F_5$.
Remark $\ $ Goldbach used the coprimality of the Fermat numbers in this way to prove that there are infinitely many primes (in a letter to Euler in 1730, see Ribenboim's The New Book of Prime Number Records, p. 4).