In this paper, we study the finiteness of the first meeting time between two different randomly moving particles in the fluid. The first particle moves with N-dimensional independent Gaussian jump and has a probability density function which is detailed in El-Hadidy (International Journal of Modern Physics B, Published Online with https://doi.org/10.1142/S0217979219502102 , 2019). The other target moves with N-dimensional independent Brownian motion. We present some analysis that proves the finiteness of the first meeting time expected value between the two particles in the fluid.