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The definition of an equivalent metric from my lecture slides:

**Two metrics: d**

_{1}and d_{2}on a metric space X are called__equivalent__when there exists M >= 1 such that M^{-1}d_{1}(x,y) <= d_{2}(x,y) <= Md_{2}(x,y)I am trying to prove this question

Show that the properties “{xn} converges to x”, “x is a cluster point of {xn}”, “{xn} is Cauchy” and “X is complete”, each remain unchanged when the metric is replaced by an equivalent metric.

However, I need to first understand the meaning behind equivalent metrics before trying to solve the problem and i've been spending an hour on it and still I don't get it. Please explain the concept behind equivalent metrics. If possible, try setting up diagrams of open/closed balls and what not to try to better understand the concept.