Note for the Small Gaps

A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted gₙ is the difference between the (n + 1)st and the nth prime numbers, i.e. gₙ=pₙ₊₁-pₙ. A twin prime is a prime that has a prime gap of two. The twin prime conjecture states that there are infinitely many twin primes. There isn't a verified solution to twin prime conjecture yet. In this note, using the Chebyshev function, we prove that n→ ∞ gₙ+gₙ₋₁log (pₙ) + log (pₙ + 2) ≥ 1, under the assumption that the twin prime conjecture is false. It is well-known the proof of Daniel Goldston, J{\'a}nos Pintz and Cem Yildirim which implies that n→ ∞ gₙlog pₙ=0. In this way, we reach an intuitive contradiction. Consequently, by reductio ad absurdum, we can conclude that the twin prime conjecture is true.