Metric proof functional analysis

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I have done part (b) as attached below and need some form of verification if I am correct or on the right track. Thanks.

Metric space definition:
M1: the distance from a point to itself is zero,
M2: the distance between two distinct points is positive,
M3: the distance from A to B is the same as the distance from B to A, and
M4: the distance from A to B (directly) is less than or equal to the distance from A to B via any third point C.