Sequences are Cauchy depending on the norm? Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am assuming that it is possible since the Cauchy test directly depends on the norm. However, this seems kind of unnatural, will someone please clarify.

It's definitely possible. Consider the real numbers with (1) the usual metric and (2) the discrete metric. Both are complete, yet the only Cauchy sequences in (2) are the eventually constant sequences, while (1) has many more Cauchy sequences.
