 and $G_\Lambda(\lambda)$ vary continuously with respect to $\Lambda$ and $\lambda$.

Now, for $\epsilon<\eta$, the minimality of $I_m$, together with the fact that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ and $\theta_0(\lambda_i)+c/2<I_m-\eta$, imply that \_(\_i)+&lt; \_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for all the points $\Lambda$ of $O\subset S$. A direct application of this fact and the Proposition 1 implies that for any point $\Lambda=$($p$, $k^\mu$) in the open set $O$, if $\gamma_\Lambda(0)=p$, the solution $\theta_\Lambda(\lambda)$ does not extend beyond a finite value $\lambda_e$ if $\epsilon<\eta$. As discussed below formula (\[asin\]), for the space time in consideration the expansion parameter $\theta_\Lambda(\lambda)$ is continuous everywhere except at the conjugate point. Therefore, the only possibility is that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$. This implies that at $\lambda_e$ there appears a conjugate point $q$ to $p$. This concludes the proof.

Proof of the second part of the third requirement
-------------------------------------------------

Before proving the second part, the following observation is needed. In the proof of Proposition \[1stpart3rdrequire\], the initial value $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$ has been selected. Choose $\eta>c/2+\epsilon$. From the minimality $I_m$ and from (\[plat\]) it follows that \_(\_i)+c&lt;\_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for every value of $\lambda\geq \lambda_i$ and $\Lambda$ in $O$. This particular choice will be useful below. In these terms, the proposition to be proved is the following.

Consider a generic point $\Lambda=(p, k^\mu)$ in an open $O\subset S$ containing $\Lambda_0=(p_0, k_0^\mu)$. Under the conditions of the Proposition \[1stpart3rdrequire\], the map $h: O\to M$ defined by $h(\Lambda)=q$, with $q$ the first conjugate point to $p$, is continuous at $\Lambda_0$.

As in the proof given before, the geodesic $\Lambda_0$ joins two conjugate points 