= Solution
As $n\to\infty$,
$$
m_n-\overline X
=\frac{\sigma^2(\mu-\overline X)}{n\nu^2+\sigma^2}
=O_p(n^{-1}),
$$
while
$$
\sqrt{v_n}
=\frac{\sigma}{\sqrt n}
\left(1+\frac{\sigma^2}{n\nu^2}\right)^{-1/2}
\sim\frac{\sigma}{\sqrt n}.
$$
Thus the credible interval and confidence interval have asymptotically equal centres and half-widths, and both contract around the true parameter. Their interpretations remain different: the confidence statement concerns repeated samples, whereas the credible statement concerns posterior probability. This is <normal-normal conjugacy with known observation variance>.
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