A Proof of the Riemann Hypothesis Based on MacLaurin Expansion of the Completed Zeta Function

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial), which can be further expressed as infinite product by conjugate complex roots. Then, according to Lemma 3, Lemma 4, and Lemma 5, the functional equation $\xi(s)=\xi(1-s)$ leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2$ with solution $\alpha_i= \frac{1}{2}$, where $\alpha_i$ are the real parts of the zeros of $\xi(s)$, i.e., $s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Thus a proof of the Riemann Hypothesis is achieved.