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    Uniqueness of the limit of a real sequence (warm-up)

    P-RA-WORKSHOP-D3-SEQ-HW1

    Area
    Real Analysis
    Topics
    Limits, Sequences of Numbers
    Status
    draft

    For a real sequence \(\{x_n\}\), if \(\lim_{n\to\infty}x_n=x\) and \(\lim_{n\to\infty}x_n=y\) then \(x=y\).

    Solution

    1. Suppose \(x_n \to x\) and \(x_n \to y\); show \(x = y\).Proof: given \(\epsilon > 0\), there are \(N_1, N_2\) with \(|x_n - x| < \epsilon/2\) for \(n \ge N_1\) and \(|x_n - y| < \epsilon/2\) for \(n \ge N_2\).For \(n \ge \max(N_1, N_2)\): \[|x - y| \le |x - x_n| + |x_n - y| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.\] Since \(\epsilon > 0\) was arbitrary, \(|x - y| = 0\), so \(x = y\).

    2. Q.E.D.

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