1 Let xnn and ynn be Cauchy sequences Prove that xn ynn is

1. Let (x_n)_n and (y_n)_n be Cauchy sequences. Prove that (x_n + y_n)_n is Cauchy.

2. Suppose that (x_n)_n is a Cauchy sequence of integers. Prove that there exists an

       N N such that for every n > N x_n = x_N.

Solution

1.Let {xn} and {yn} be Cauchy.

Then for any > 0 there exists N1 N such that for any n, m N1, |xn xm| < 2 ,

and there exists N2 N such that for any n, m N2, |yn ym| < 2 .

Let N = max(N1, N2).

Then for any n, m N, |(xn + yn) (xm + ym)| = |(xn xm) + (yn ym)| |xn xm| + |yn ym| < 2 + 2 = . Thus {xn + yn} is Cauchy.