I am reading Hardy's proof that the Riemann zeta function has infinitely many zeros on the critical line. He uses the fact that if $k>\frac{1}{2}$ and $Re(y)>0$, then
$\frac{1}{\pi i} \int_{k-i \infty}^{k+i \infty} \Gamma(u) y^{-u} \zeta(2u) du = \sqrt{\frac{\pi}{y}}-\frac{1}{\pi i} \int_{\frac{1}{4}-i \infty}^{\frac{1}{4}+i \infty} \Gamma(u) y^{-u} \zeta(2u) du$
Why is this true? This would follow from the residue theorem if the top and bottom edges of the rectangle contour had contribution tending to zero. How to prove that they do?