\frac 1 2<\frac 1 2+\frac i{2\cdot\left(2\left\lceil\frac t{\sqrt 2}\right\rceil+1\right)}\leq \frac 1 2+\frac{2\left\lceil\frac t{\sqrt 2}\right\rceil}{2\cdot\left(2\left\lceil\frac t{\sqrt 2}\right\rceil+1\right)}<\frac 1 2+\frac 1 2=1