$\displaystyle \exists t \geq 0 \forall s \geq 0,$ we have $\displaystyle s \geq t$

$\displaystyle t = 0$ works.

$\displaystyle \exists s \geq 0 \forall t \geq 0,$ we have $\displaystyle s \geq t$

Why does $\displaystyle s = t+1$ not work? The book said

$\displaystyle \exists s \geq 0 \forall t \geq 0,$ we have $\displaystyle s \geq t$ is false since its negation is true. But I would like to know why could s = t+1 not be used?

Thanks