For each positive integer k, define for . Is the sequence a Cauchy sequence in the metric space ?
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;
so when x=0.
Case 2: 0 < x < 1;
so when 0 < x < 1.
Case 3: x=1;
so when x = 1.
Therefore converges in , thus proves it is a Cauchy sequence.
Is this convincing? Thanks.