Note for the Small Gaps

A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted g₍ is the difference between the (n + 1) st and the nth prime numbers, i. e. g₍=p₍+₁-p₍. 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 ₍ {g₍+g₍-₁ (p₍) + (p₍ + 2) } 1, under the assumption that the twin prime conjecture is false. It is well-known the proof of Daniel Goldston, J\'anos Pintz and Cem Yildirim which implies that ₍ {g₍ p₍}=0. In this way, we reach an intuitive contradiction. Consequently, by reductio ad absurdum, we can conclude that the twin prime conjecture is true.