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?

• I think the limit $\lim_{s \to 1} \zeta(s)$ does not exists. – Botond Aug 23 '19 at 12:29
• It's not right because the limits you refer to don't exist, so you can't use the usual theorems on them – Maxime Ramzi Aug 23 '19 at 12:29
• To emphasize: IF $\lim_{n \to \infty} x_n = L$ AND $\lim_{n \to \infty} y_n = M$ THEN $\lim_{n \to \infty} (x_n + y_n) = L + M$. If you cannot verify the two hypotheses, then you are not justified in using the conclusion. – Lee Mosher Aug 23 '19 at 12:35
• How do you define $\zeta (1)$ – reuns Aug 23 '19 at 12:39
• The Cauchy' sprincipal value of $\zeta(1)$ is indeed $\gamma$. – Anixx Aug 24 '19 at 5:04

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).
• E.g., $\lim n+\lim(7-n)=7$. – Gerry Myerson Aug 23 '19 at 12:32