\sum_{i=1}^{2\left\lceil\frac t{\sqrt 2}\right\rceil}\sqrt 2\cdot\left(\frac 1 2+\frac i{2\cdot\left(2\left\lceil\frac t{\sqrt 2}\right\rceil+1\right)}\right)>\sum_{i=1}^{2\left\lceil\frac t{\sqrt 2}\right\rceil}\frac{\sqrt 2}2=2\left\lceil\frac t{\sqrt 2}\right\rceil\cdot\frac{\sqrt 2}2=\sqrt 2\left\lceil\frac t{\sqrt 2}\right\rceil\geq\sqrt 2\cdot\frac t{\sqrt 2}=t