Hi all, does anyone know the following proofs?

Suppose that you have a random sample x1, x2, . . . , xn from an N(μ,1/r) distri-
bution (with r known) and take a N(b,1/d) prior for μ. Verify that:

(i) the posterior mean is greater than the prior mean if and only if the likelihood
mode is greater than the prior mean;

(ii) the standard deviation of the posterior distribution is smaller than that of
the prior distribution.

Thanks!