ANALYSIS AND PROOF OF THE RIEMANN HYPOTHESIS

We propose a proof for the Riemann Hypothesis by dividing the Dirichlet eta function into a main term and a remainder term, focusing primarily on the behavior of the remainder in the critical strip \textit{(}$0<\sigma<1$ \textit{where }$s=\sigma+it$\textit{)}. Then, we express the Riemann zeta function using the same decomposition and show that its main term cannot vanish at the nontrivial zeros. Finally, we focus on the limit on the main terms as $\left\vert \lim k\rightarrow \mathbf{\infty}\text{ \ }\zeta_{k}(s_{0})/\zeta_{k}(1-s_{0})\right\vert $.