The following statements can be found at Glasserman's Monte Carlo Methods in Financial Engineering.
Given a stochastic process $(X_t)_{t\in [0,T]},$ if
$(i)$ the mapping $t\mapsto X_t$ is continuous almost everywhere on $[0,T],$ and
$(ii)$ $X_t$ has independent increments (ie for any $\{0\leq t_0<t_1<\dots <t_n\leq T\}$, all increments $\{ W(t_n) - W(t_{n-1}), W(t_{n-1}) - W(t_{n-2}),...,W(t_1)-W(t_0)\}$ are independent.) and $X_t - X_s$ has mean $0$ and variance $t - s$ for all $s<t,$
then $X_t - X_s$ follows a normal distribution.
When I tried to show that $X_t- X_s$ follows a normal distribution, I tried to show that its MGF is the same as a normal distribution, that is, if $Y = X_t - X_s$, then $$\mathbb{E} [e^{uY}] = e^{\frac{1}{2} u^2(t-s)^2}.$$ However, I am not able to do it. Any hint is appreciated.