A strictly monotonically increasing sequence of transcendental numbers is constructed, which converges to the Euler-Mascheroni constant. This proves that the constant is transcendental.

The basic tool is about Bessel functions and there related Mellin transforms in combination with the related technique of R. P. Brent regarding the "asymptotic expansions inspired by Ramanujan", (BrR).

  

Braun K., A Bessel function based proof that the Euler constant is irrational