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Dirichlet approximation

  1. (Trivial approximation) For \(x \in \mathbb{R}\) and \(n \in \mathbb{Z}^+\), there is a rational number \(\frac{p}{q}\), with \(1 \leq q \leq n\), such that

    \(\left|x - \frac{p}{q}\right| < \frac{1}{n}\)

  2. (Dirichlet approximation) For \(x \in \mathbb{R}\) and \(n \in \mathbb{Z}^+\), there is a rational number \(\frac{p}{q}\), with \(1 \leq q \leq n\), such that

    \(\left|x - \frac{p}{q}\right| < \frac{1}{nq}\)

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