 Solution

The requested conclusion is false for the printed parameter values. They give $V_0=1.01/n$ and $V_1=101/n$. Substituting $y=3/\sqrt{n}$ in the Bayes factor yieldsThus the Bayes factor favours $H_1$ by about $8.23$ to one, not $H_0$. Nor does the conclusion hold for arbitrary large $n_0,c$: both parameters enter the expression explicitly. With $n_0/n=100$, taking a much wider alternative, for example $c=1000$, would instead give $B_{01}\simeq 1.1563>1$. That is a different prior assumption and cannot repair the printed calculation silently.