= Solution
Write $\sigma^2=\Sigma_\eta>0$ and assume $V_0>0$. The scalar <Kalman filter> reduces to
$$
V_{t+1}=\frac{V_t\sigma^2}{V_t+\sigma^2},\qquad
\hat X_{t+1}=\frac{\sigma^2\hat X_t+V_tY_{t+1}}{V_t+\sigma^2}.
$$
Taking reciprocal variance and dividing the mean by variance gives
$$
V_{t+1}^{-1}=V_t^{-1}+\sigma^{-2},\qquad
\hat X_{t+1}/V_{t+1}=\hat X_t/V_t+Y_{t+1}/\sigma^2.
$$
Thus
$$
\hat X_t=\frac{\sigma^2\hat X_0/V_0+\sum_{j=1}^tY_j}{\sigma^2/V_0+t}.
$$
Since the process noise is zero and $A=1$, $X_t=X_0$ and $Y_j=X_0+\eta_j$. The <strong law of large numbers> gives $t^{-1}\sum_jY_j\to X_0$ almost surely. The prior's finite weight vanishes in the displayed formula, so \b[both limits equal $X_0$ almost surely.] A zero prior variance instead means $X_0$ is known, and the assertion follows directly without reciprocal-variance formulas.
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