Riemann zeta function (Edwards) Section 2.4

Question

In section 2.4 of the book "Riemann's zeta function" by Edwards, the estimate $n(R)\leq 2R\log(R)$ is derived, where $n(R)$ is the number of roots of the equation $\xi(\rho)=0$ inside the circle $|s-1/2|=R$. But the last line of the proof is $$n(R)\leq \frac{2}{\log 2}R \log R +2R -\frac{\xi(1/2)}{\log 2}\leq 2R \log R$$ and the second inequality is not true for large $R$. It would become true by replacing $2R\log R$ with $3R\log R$. Is this a typo in the book, or am I missing something?