Asymptotic Mean of Tn

Question

So suppose that:

$$ \sqrt n(Tn - \theta) \rightarrow d\rightarrow T \sim {N}(0,\varepsilon) $$

and Tn is asymptotically normal.

How do I find the asymptotical mean of Tn?

So far I have:

$$ (Tn - \theta) \approx \frac{Tn}{\sqrt n} $$

$$ Tn \approx \frac{Tn}{\sqrt n} + \theta $$

What's the next step? I know the answer is $\theta$, but I don't know why or how.

Answer 1

For $n\rightarrow \infty$, what happens to $T_n$?