Definition. Let $M$ be a regular $C^2$-surface in $\mathbb{R}^3$, $p\in M$ and $$T_p^1M=\{e\in T_pM:\lVert e\rVert=1\}.$$ be the unit circle in the tangent plane $T_pM$. Then every non-zero tangent vector $Z\in T_pM$ can be written as $$Z=r_Z\cdot e_Z,$$ where $r_Z=\lVert Z\rVert$ and $e_Z=\frac{Z}{\lVert Z\rVert}\in T_p^1M$. For a unit tangent vector $e\in T_0^1M$ let $$\gamma_e:(-a_e,b_e)\to M$$ be the unique maximal geodesic such that $a_e,b_e\in\mathbb{R}^+\cup\{\infty\}$, $\gamma_e(0)=p$, $\gamma'_e(0)=e$. It can be shown that the real number $$\varepsilon_p=\inf\{a_e,b_e:e\in T_p^1M\}$$ is positive, so the open ball $$B_{\varepsilon_p}^2(0)=\{Z\in T_pM:\lVert Z\rVert<\varepsilon_p\}$$ is non-empty. Then the exponential map $\exp_p:B_{\varepsilon_p}^2(0)\to M$ at $p$ is defined by $$\exp_p(Z)=\begin{cases} p,& Z=0,\\ \gamma_{e_Z}(r_Z),& Z\neq 0. \end{cases}$$
I have been trying to understand the proof that shows that geodesics are locally the shortest paths between their endpoints. And I get most of it but I do not understand one point of the definition of the exponential map we use here:
Let $\gamma : (-a,b) \to M$ be the unique maximal geodesic such that $a,b$ are strictly positive or infinite, with $\gamma$ parametrized by arclength, $\gamma(0)=p$ and $\gamma'(0)=e$.
Why can we say that $\varepsilon = \inf\{a,b\}$ is going to be strictly positive?
I have been having a really hard time understanding this because for example : Take $M$ the unit sphere and $p=(1,0,0)$. One could take $a$ to be a little bit bigger than $0$ and $b=2\pi-a$ and then one would get $\varepsilon=\inf\{a\}=0$.
Is there something I am misunderstanding or completely missing here?
I would be happy to get any help. Thank you in advance.