Is my proof that $\gamma$ (the Euler-Mascheroni constant) is transcendental correct?

Question

The Euler-Mascheroni constant $\gamma$ can be defined as $\lim\limits_{n\to \infty}(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}-\ln n)$. For every positive integer n (except for 1), the value of this sequence is transcendental. So from the definition, $\gamma$ must be transcendental, too.

Answer 1

No.

$\lim_{n\to\infty}\frac\pi n$ is the limit of transcendental numbers and yet rational

Answer 2

Rationals are dense so find $a_n\rightarrow e$ and $a_n\in\mathbb{Q}$. Then define $$b_n=a_n-e.$$