I am reading the book "A course in Minimal Surfaces" by Colding and Minicozzi. I don't understand a step in the proof of Corollary 1.13.
Let $\Sigma^k \subset \mathbb{R}^n$ be a $k$-dimensional minimal submanifold. Fix $x_0 \in \Sigma$. I want to study the behaviour of the function: $$ s\mapsto \Theta_{x_0}(s) \,\, \colon = \frac{ \text{Vol}\big( B_s(x_0) \cap \Sigma \big) }{\text{Vol}\big(B_s \subset \mathbb{R}^k \big)} $$ where $B_s(x_0)$ is the $n$-dimensional euclidean ball of radius $s$ centred in $x_0$ and $B_s$ is the $k$-dimensional euclidean ball of radius $s$ centred in the origin.
I know that $\Theta_{x_0}(s)$ is monotone nondecreasing.
I want to show that $$ \Theta_{x_0}(s) \ge 1. $$ In the book, the authors say that since $\Sigma$ is smooth and proper, it is infinitesimally Euclidean and hence $$ \lim_{s \rightarrow 0}\Theta_{x_0}(s) \ge 1. $$
Can you explain me this better?
If $\Sigma$ is smooth and proper then for $x_0\in \Sigma$, we know that $\exists$ a small ball in $\mathbb{R^k}$ around $x_0$ (by the definition of a manifold). But the ball can intersect $M$ multiple times (but it intersect $M$ at least once).
So the numerator in the definition of $\Theta$ is the number of times the ball intersects $M$ and hence $\lim_{s\to 0}\Theta_{x_0}(s)\geq 1$.