# Confusion about the delta method

I'm reading Statistical Models by A. C. Davison and I'm really confused by this section on the Delta method.

1. It's not mentioned explicitly, but is $$h(T_n)$$ a consistent estimator of $$h(\mu)$$?

2. In the Taylor series expansion, I'm not sure why the term inside $$h'$$ is $$\mu+n^{-1/2}\tau W_n$$ if the expansion is around $$\mu$$, and why the expansion is an inequality when higher order derivatives are omitted.

3. Why can $$\frac{n^{1/2}(h(T_n)-h(\mu))}{\tau h'(\mu+n^{-1/2}\tau W_n)}$$ be substituted with $$Z_n$$?

4. From the last line, why does $$h(T_n)$$ have expected value $$h(\mu)$$ or is the relationship approximate/asymptotic? By Jensen's inequality, if $$E[T_n] = \mu$$, I don't think $$E[h(T_n)]$$ equals $$E[h(\mu)]$$?