The Penrose singularity theorem (1965) proved that once a trapped surface forms — a 2-surface where even outgoing light rays converge — a singularity is inevitable under the null energy condition. The key ingredient is the Raychaudhuri equation: dθ/dλ = -θ²/2 - σ²+ ω² - R_{μν}k^μk^ν, showing that positive energy causes focusing (θ decreasing), inevitably reaching θ→-∞ (a caustic = singularity) in finite affine parameter. Hawking extended this to cosmology, proving the Big Bang singularity is also inevitable. Together the Hawking-Penrose theorems showed GR necessarily predicts its own breakdown.