Is the cube of a Cauchy sequence of real numbers Cauchy?

Question

I am thinking yes, because a Cauchy sequence converges, so we can use the limit law for products twice, declare the cube of the sequence convergent, implying the cube is Cauchy. Is this correct? Are there general principles I am not aware of that would apply in wider circumstances?

Answer 1

Your reasoning is correct. A Cauchy sequence in the real numbers converges by completeness. The cube of a convergent sequence converges, and convergent sequences are Cauchy.