Small gaps between primes or almost primes

Let $p_n$ denote the $n^{{\rm th}}$ prime. Goldston, Pintz, and Yıldırım recently proved that

$$\liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$$

We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{{\rm th}}$ number that is a product of exactly two distinct primes. We prove that

$$\liminf_{n\to \infty} (q_{n+1}-q_n) \le 26.$$

If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to $6$.