Fake proof that $\zeta (1)=\gamma$ The following proof is fake, but I have no idea why it's not right (I know it's paradoxical, since $\zeta (1)$ diverges): $$\begin{align*}\zeta (1)&=\dfrac{2\zeta (1)}{2}\\\&=\dfrac{\displaystyle\lim_{x\to 0}\zeta (1+x)+\displaystyle\lim_{x\to 0}\zeta (1-x)}{2}\\\&\overset{!}{=}\displaystyle\lim_{x\to 0}\dfrac{\zeta (1+x)+\zeta (1-x)}{2}\\\&=\gamma\approx 0.577.\end{align*}$$ Tannery's theorem is of the form $a\implies b$. Here, I did it the other way around: $b\implies a$. So, that's not equivalent (but is close) to Tannery's theorem. Where is the mistake? I suspect it's the "$!$", but which theorem does it refer to?

You cannot use the rule $\lim\limits_{n \to \infty} (a_n + b_n) = \lim\limits_{n \to \infty} a_n + \lim\limits_{n \to \infty} b_n$ here because the limits do not exist. Like that you could create all kinds of wrong results ($\infty - \infty$ can be anything).