Verified answer

Let ε > 0 be given.

Since f is uniformly continuous on I and {x_n} ⊆ I, there exists δ > 0 such that

|x_m - x_n| < δ ==> |f(x_m) - f(x_n)| < ε.

Next, since {x_n} is Cauchy, there exists a positive integer N such that

|x_m - x_n| < δ for all m, n > N.

Hence, |f(x_m) - f(x_n)| < ε for all m, n > N.

==> {f(x_n)} is Cauchy.

I hope this helps!