The definition of the Euler–Mascheroni constant is the limit of $$H_n - \log(n)$$ as n approaches infinity. So, why is it so hard to prove the irrationality of this constant? $H_n$ is defined only for integers and for any integer $n > 1$ , $\log(n)$ is irrational.
On the other hand, $H_n$ is always a rational number. Subtracting an irrational number from a rational one doesn't make the constant an irrational number?
Consider the sequence $(x_n)_{n\in\mathbb N}$ defined by$$x_n=\log(2)-\sum_{k=1}^n\frac{(-1)^{k+1}}k.$$Each $x_n$ is an irrational number minus a rational number. However, $\lim_{n\to\infty}x_n=0$ and $0$ is rational.