you can find the definition of a Wiener process at the the following link :
http://en.wikipedia.org/wiki/Wiener_process, in the following discussion we will escape from the rigorous mathematical framework in order to extract the main idea of the proof
it says that W(t) has independent increment : W(t) – W(s) ~ N(0 , t-s) for 0 ≤ t < s where N is the famous normal distribution : http://en.wikipedia.org/wiki/Normal_distribution
then we can rewrite for a small increment of t : (*) W(t + Δt ) – W(t) = Φ √Δt, where Φ is a random variable which follows N(0,1)
Therefore at time t , (meaning t is fixed, the variable is Δt ) E[ W(t + Δt) – W(t)] = 0
then E[ W( t + Δt)] = W(t)
we can write for all ε > 0 Prob( W(t+ Δt) – W(t) > ε) = Prob( W(t+ Δt) – E[ W( t + Δt)] > ε), we can use the Chebyshev inequality
Prob( W(t+ Δt) – E[ W( t + Δt)] > ε) ≤ var[ W(t+ Δt) ] / ε*ε
from (*) we got var[ W(t+ Δt) ] = Δt
Finally Prob( W(t+ Δt) – E[ W( t + Δt)] > ε) ≤ Δt / ε*ε
then for all ε > 0 Prob( W(t+ Δt) – E[ W( t + Δt)] > ε) = 0 when Δt → 0
