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The inequality in line one follows from repeated applications of the [[triangle inequality]]; the series in line four is a [[geometric series]] with <math>0 \leq q < 1</math> and hence it converges. The above shows that <math>\{x_n\}_{n\geq 0}</math> is a [[Cauchy sequence]] in <math>(X, d)\,</math> and hence convergent by completeness. So let <math>x^* = \lim_{n\to\infty} x_n</math>. We make two claims: (1) <math>x^*\,</math> is a [[fixed point]] of <math>T\,</math>. That is, <math>Tx^* = x^*\,</math>; (2) <math>x^*\,</math> is the only fixed point of <math>T\,</math> in <math>(X, d)\,</math>.
To see (1), we note that for any <math>n \in \{0, 1, \ldots\}</math>,
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