= Solution
One standard form of the <Penrose singularity theorem> assumes a time-oriented <globally hyperbolic spacetime> with a noncompact <Cauchy hypersurface>, the <null convergence condition> $R_{ab}k^ak^b\geq0$ for every <null vector> $k$, and a closed future <trapped surface>. It concludes \b[future null geodesic incompleteness]: some future-inextendible <null geodesic> has a finite upper endpoint of its <affine parameter>.
The focusing mechanism is the <Null Raychaudhuri equation>. The normal generators have zero <null twist>, so
$$
\frac{d\theta}{d\lambda}=-\frac12\theta^2-\sigma_{ab}\sigma^{ab}-R_{ab}k^ak^b\leq-\frac12\theta^2.
$$
An initial $\theta_0<0$ therefore gives a <conjugate point to a spacelike surface> within affine distance at most $2/|\theta_0|$, assuming the generator can be continued that far. Such a generator ceases to lie on the <achronal boundary> after its first focal point. Compactness of the <trapped surface> supplies a uniform bound for all normalized initial null normals. Future completeness would consequently make its future boundary compact. Projection along timelike curves to a connected <Cauchy hypersurface> is injective on the <achronal boundary> and has open image. Compactness makes the image closed as well; the nonempty image must therefore be the entire hypersurface, contradicting its noncompactness. This explains why the global assumptions supplement local focusing.
For <Reissner-Nordstrom spacetime>, the Maxwell <stress-energy tensor> satisfies the <null energy condition>; the <Einstein field equations> imply the required <null convergence condition>. The spheres in the band just found are closed and trapped. Apply the theorem to a <maximal Cauchy development> with a noncompact <Cauchy hypersurface> and containing one such sphere. That <globally hyperbolic spacetime> must be future <null-geodesically incomplete>.
The <Penrose theorem at a Cauchy horizon> needs care: the full maximal analytic <Reissner-Nordstrom spacetime> has inner <Cauchy horizons> and is not globally hyperbolic. It does not satisfy every hypothesis of the displayed theorem. In the exact solution some incomplete geodesics of the globally hyperbolic development reach a smoothly extendible <Cauchy horizon> in finite <affine parameter>. The theorem asserts incompleteness of the development, not that every such endpoint is a curvature singularity. The separate curvature singularity at $r=0$ does not justify silently dropping the theorem's global hypothesis.
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