 Solution

Suppose inductively thatThe observation innovation isso its conditional variance is $V_{t-1}+1$ and its conditional covariance with $x_{t-1}$ is $V_{t-1}$. The scalar static-state Kalman update therefore givesAdding the known control $u_{t-1}$ shifts the conditional mean but not the variance, giving exactlySince $V_0=1$, induction yieldsEvery update is an affine conditioning operation on jointly Gaussian random variables, so