It's close to trivial! The only place t appears on the right is in the numerator of a fraction so, as t goes to infinity that fraction goes to infinity, but that means the exponent is going to negative infinity so the exponential goes to 0.
On the left, a sum CAN'T be greater than infinity! So the probability on the left is also 0.
As t goes to 0, that inequality reduces to "0= 0".
