Metric in normal coordinate

Question

Using Gauss’s lemma we can write the metric in normal co-ordinate as $g(r, θ) = dr^2 + r^2h_{ij}(r, θ)dθ^i ⊗ dθ^j$ (where metric on $S^{n-1}$ is $\tilde {g}=dθ^i ⊗ dθ^i$). Now as $r \rightarrow 0$, $g$ tends to Euclidean metric. My feeling is that this implies $\lim _{r \rightarrow 0}h_{ij}(r, θ)=\delta_{ij}$. Am I correct?