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Metric Spaces

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Give an example of each of the following:

a) an infinite subset of R with no cluster point
b) a complete metric space that is bounded but not compact
c) a metric space none of whose closed balls is complete

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a) We consider the integer set Z in R. Z has no cluster point.
Because for any integer n in Z, we consider its neighborhood (n - 0.5, n + 0.5). In this neighborhood, there does not exist another point in Z.

b) For the real set R, we consider the discrete metric d with d(x, y) = 1 if x is not equal to y and d(x, x) = 0.
Then R is ...

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This solution explains how to solve the given problems involving Metric Spaces.

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Metric Space Proofs

Problem 1: Given the metric space (X, p), prove that

a) |p(x, z) - p(y, u)| < p(x, y) + p(z, u) (x, y, z, u is an element of X);

b) |p(x, y) - p(y, z)| < p(x, y) (x, y, z is an element of X).

These problems are from Metric Space. Please give formal proofs for both (a) and (b) based on the reference provided. Thank you.

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