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Convergence = unique limit

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Show that a convergent sequence in a metric space has a unique limit.

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This solution is comprised of a detailed explanation to show that a convergent sequence in a metric space has a unique limit.

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Solution. Let's consider all of the possible cases we can have in a 3 × 3 system.
1) Success. In order for a system to be successful through elimination, there may be two possibilities for the structure of the coefficient matrix.

a) Full Success. The matrix may contain three independent columns. This way, the columns span all of space, and the planes formed by the rows intersect at a single point. Example:

b) Temporary Failure. The matrix may need a row exchange to allow for a pivot to be found. After this, three ...

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