proving this inequality $\left | \int_{\left | z \right |=2}^{}\frac{dz}{z^2+1} \right |\leq \frac{4\pi}{3}$ proving this inequality $$\left | \int_{\left | z \right |=2}^{}\frac{dz}{z^2+1} \right |\leq \frac{4\pi}{3}$$
I tried with $$\left | \int_{\left | z \right |=2}^{}\frac{dz}{z^2+1} \right |\leq \int_{\left | z \right |=2}^{} \left | \frac{dz}{z^2+1} \right |$$ 
but I dont know what next 
I think maybe we need to use triangular inequality 

$$\left|\;\;\oint\limits_{|z|=2}\frac{dz}{z^2+1}\;\;\right|\le\oint\limits_{|z|=2}\frac{dz}{|z^2+1|}\stackrel{\text{Est. Lem.}}\le\max_{|z|=2}\left(\frac1{|z|^2-1}\right)\cdot l\left(\{|z|=2\}\right)=$$
$$=\frac13\cdot4\pi=\frac{4\pi}3$$
Est.Lem. = Estimation Lemma of Cauchy
