We consider the alternating Riemann zeta function $\zeta^*(s)= \sum^{\infty}_{ n=1} \frac{(-1)^{n-1}}{n^s}$, which converges if $Re (s)>0 .$ By using Rouche's theorem, Vitali's theorem and by method of contradiction we complete the proof of the Riemann Hypothesis.

General Mathematics 11M06, 11M41 F.2.2

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