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?
I love testing a device and feeling like the manufacturer has thought of everything -- but that's rare in the smart home world. Manufacturers are challenged to balance performance, build quality, experience, and affordability, while also trying to make a profit. As a result, corners can be cut, features downgraded, and compromises made.
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