To prove this lemma, we need to rescale both the metric and the coordinates. We choose the pull-back region $\pi^{-1}(B_{r_{0}}^{g_{0}}(p_{m}))$ with $r_{0}>0$ defined as the above, then for every $\bm{x}\in\pi^{-1}(B_{r_{0}}^{g_{0}}(p_{m}))$ with $\pi(\bm{x})=(x,y,z)$,

(7.55)

$$V_{\beta}(\bm{x})=\frac{1}{2\sqrt{x^{2}+y^{2}+z^{2}}}+h(\bm{x})+\beta,$$

where $h$ is a bounded harmonic function on $\mathbb{R}^{3}$. Let us denote the rescaled coordinates by

(7.56)

$\displaystyle\tilde{x}_{\beta}\equiv\gamma_{\beta}\cdot x,\ \tilde{y}_{\beta}\equiv\gamma_{\beta}\cdot y,\ \tilde{z}_{\beta}\equiv\gamma_{\beta}\cdot z,$

and we choose

(7.57)

$$\gamma_{\beta}\equiv\sigma^{2}\cdot\beta.$$

So the rescaled metrics $\tilde{g}_{\sigma,\beta}$ converge to

(7.58)

$$\tilde{g}_{\sigma,\infty}=G_{\sigma}\cdot g_{\mathbb{R}^{3}}+(G_{\sigma})^{-1}\theta^{2}$$

such that

(7.59)

$$G_{\sigma}(p)=\frac{1}{2d_{0}(p,0^{3})}+\frac{1}{\sigma^{2}},$$

where $d_{0}$ is the distance function in the Euclidean space $\mathbb{R}^{3}$. This tells us that $\tilde{g}_{\sigma,\infty}$ is a Taub-NUT metric, and the proof is complete.

∎