  The Dirichlet eta function can be divided into $n$-th partial sum
$\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term
which can be approximated by the expression for $n$. And then, to increase
reliability, we make sure that the error between remainder term and its
approximation is reduced as n goes to infinity. According to the Riemann zeta
functional equation, if $\eta(\sigma+it)=0$ then $\eta(1-\sigma-it)=0$. In this
case, $n$-th partial sum also can be approximated by expression for $n$. Based
on this approximation, we prove the Riemann hypothesis.
