Show directly that any multiple of a Cauchy sequence is again a Cauchy sequence

Suppose {x_n} is Cauchy and c is a real constant.
We need to show that {c*x_n} is Cauchy.

If c = 0, then {c*x_n} is the zero sequence, which is trivially Cauchy since the absolute value of the difference of any two terms of the zero sequence is zero (which is less than any positive epsilon).

Now suppose c is nonzero, and let epsilon > 0. Since {x_n} is Cauchy, there exists a natural number N such that for all m, n > N,

|x_n - x_m| < epsilon/|c|.

Therefore, for all m, n > N,

|c*x_n - c*x_m| = |c(x_n-x_m)| = |c||x_n-x_m| < |c|(epsilon/|c|) = epsilon.

So {c*x_n} is Cauchy.

We conclude that any multiple of a Cauchy sequence is again a Cauchy sequence.

Lord bless you today!