Stationarity

Hello

I would appreciate to know if this is true for a Winer process:

For t around 0 (Δ𝑡→0)
〖𝑊_𝑡=𝑊_𝑡 −𝑊_0≡𝑑𝑊〗_𝑡 ∼𝒩(0,𝑑𝑡)≡𝒩(0,𝑡−0)≡𝒩(0,𝑡)

For any 𝑠<𝜏 (Δ𝑠=(𝜏−𝑠)→0)
𝑊_𝜏 −𝑊_𝑠≡〖𝑑𝑊〗_𝑠 ∼𝒩(0,𝑑𝑠)≡𝒩(0,𝜏−𝑠)
𝑊_𝜏 −𝑊_𝑠∼𝒩(0,𝜏)−𝒩(0,𝑠)=𝒩(0,𝜏−𝑠)
𝑊_𝜏 −𝑊_𝑠≡〖𝑑𝑊〗_𝑠=𝑊_𝑑𝑠=𝑊_(𝜏−𝑠)