= Solution
<Banach fixed-point theorem>[Banach's fixed-point theorem] states that a map $T:Y\to Y$ on a nonempty <complete metric space> with $d(Tx,Ty)\le qd(x,y)$ for a constant $0\le q<1$ has a unique <fixed point>, approached from every starting point by iteration.
Choose $x_0$ and let $x_{n+1}=Tx_n$. Induction gives $d(x_{n+1},x_n)\le q^nd(x_1,x_0)$. For $m>n$, the triangle inequality and <geometric series> give
$$
d(x_m,x_n)\le\frac{q^n}{1-q}d(x_1,x_0).
$$
Thus the <sequence> is Cauchy and converges by completeness to some $x_*$. The contraction estimate implies <continuity>, so $Tx_*=\lim Tx_n=\lim x_{n+1}=x_*$. If $y_*$ were another <fixed point>, $d(x_*,y_*)\le qd(x_*,y_*)$ would force equality of the points. Passing $m\to\infty$ also yields the quantitative error estimate
$$
\boxed{d(x_n,x_*)\le\frac{q^n}{1-q}d(x_1,x_0).}
$$
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