With Theorem 5.3 in hand, the proof of Theorem 1.1 becomes standard, and follows the same lines as the proof of Corollary 5.2. Thus, consider a sequence

$\displaystyle(M^{n}_{j},d_{j},p_{j})\to(X,d,p)$

(5.11)

of Riemannian manifolds satisfying $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ and ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$, which Gromov-Hausdorff converge to some $X$. Recall again the standard stratification of $X$, which is reviewed in Section 2.1. More specifically let us consider the closed stratum $\mathcal{S}^{n-4}(X)\subseteq X$. On the one hand, we have from [ChCo1]

$\displaystyle\dim\mathcal{S}^{n-4}\leq n-4\,.$

(5.12)

On the other hand, we have that for every point $x\not\in\mathcal{S}^{n-4}$, there exists some tangent cone at $x$ which is isometric to $\mathds{R}^{n-3}\times C(Y)$. That is, for some sequence $r_{a}\to 0$ we have

$\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n-3}\times C(Y)\,.$

(5.13)

However, by Theorem 5.3, we have that $Y$ is isometric to the unit $2$-sphere, and hence,

$\displaystyle(X,r_{a}^{-1}d,x)\to\mathds{R}^{n}\,.$

(5.14)

Then for $a\in\mathds{N}$ sufficiently large, we can apply the standard $\epsilon$-regularity theorem, Theorem 2.3, to see that $r_{h}(x)>0$, and hence that a neighborhood of $x$ is a $C^{1,\alpha}$ Riemannian manifold, which proves the theorem.