= Solution
The <contraction mapping theorem> states that a <contraction mapping> of a \b[nonempty complete metric space] into itself has a unique <fixed point> $x_*$. For any starting point $x_0$, the iterates $x_{n+1}=f(x_n)$ converge to $x_*$, with the quantitative estimate
$$
\boxed{d(x_n,x_*)\leq\frac{q^n}{1-q}d(x_1,x_0).}
$$
Completeness and the self-map condition are essential: the geometric estimate makes the iterates a <Cauchy sequence>, completeness supplies their limit in $X$, and continuity of the <contraction mapping> makes that limit a <fixed point>. If two <fixed points> existed, their distance would be at most $q$ times itself, forcing equality of the points.
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