Equivalent metrics are two metrics that induce the same topology on a set. In other words, they define the same notions of open and closed sets, and therefore have the same convergent sequences.
To prove two metrics are equivalent, you must show that every open set in one metric is also an open set in the other metric, and vice versa. This can be done by showing that any sequence that is convergent in one metric is also convergent in the other metric.
Yes, it is possible for two metrics to be equivalent on one set but not on another. This is because the definition of equivalent metrics depends on the topology induced on the set, which can vary from set to set.
Equivalent metrics do not affect the convergence of sequences. This means that if a sequence is convergent in one metric, it will also be convergent in an equivalent metric, and vice versa.
Yes, all convergent sequences in one metric are also convergent in an equivalent metric. This is because equivalent metrics define the same notion of convergence, so any sequence that is convergent in one metric will also be convergent in the other.