A vanishing $c/\sqrt n$ shift keeps consistency but moves the normal limit to $N(c,1)$

Let $T_n$ be a consistent estimator of $\theta$ such that $\sqrt n(T_n-\theta)\to N(0,1)$ in distribution and $T_n\overset{P}{\longrightarrow}\theta$. Let $T_n^*$ be an estimator of $\theta$ given by $T_n^*=T_n+\dfrac c{\sqrt n}$, $c>0$. Then: