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\end{align*}  Now using Proposition 4.8(d) and the reverse triangle inequality,  \begin{align*}  \lim_{R\rightarrow\infty}\left|\int_{\gamma_R}\frac{dz}{z^{4}+1}\right| \left|\int_{\gamma_R}\frac{dz}{z^{4}+1}\right|  &\leq max_{z\in\gamma_R}\left|\frac{1}{z^{4}+1}\right|\pi R\\ &\leq max_{z\in\gamma_R}\left(\frac{1}{|z|^{4}-1}\right( max_{z\in\gamma_R}\left(\frac{1}{|z|^{4}-1}\right) = \frac{\pi R}{R^{4}-1}  \end{align*}