For each positive integer k, define for . Is the sequence a Cauchy sequence in the metric space ?

My proof:

Since is continuous, it is in .

Now, for each E > 0, there exists N in the set of Natural numbers such that k >= N, we have:

Case 1: x=0;

then ,

so when x=0.

Case 2: 0 < x < 1;

then ,

so when 0 < x < 1.

Case 3: x=1;

then ,

so when x = 1.

Therefore converges in , thus proves it is a Cauchy sequence.

Q.E.D.

Is this convincing? Thanks.