This proof is known as Dirichlet’s approximation theorem. At first blush, it only guarantees a single 2-good approximation for every real. Worse yet, the solution guaranteed by the proof might not comply with the rules of our game, because nothing stops it from producing exact approximations (ε = 0). So, what did we achieve?
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Washington endorsed Pakistan’s “right to defend itself” after it bombed major cities across Afghanistan amid heightened tensions between the two hostile neighbours.