Let be an isometry in a metric space . Suppose

Could you explain to me why the sequence is monotone and why does it converge with respect to the usual topology of and also with respect to and, in particular, it satisfies the condition of Cauchy?

I know that since isometry is a continuous and bijective function, it must be monotone. We put , so it must be increasing. But I can't figure out why

Could you help me with that? I would really appreciate a thorough explanation, because I can't find anything about isometry composition anywhere.

Thank you.