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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|\leq \lim_{R\rightarrow\infty}\left|\int_{\gamma_R}  \end{align*}