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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}\right| \lim_{R\rightarrow\infty}\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  \end{align*}