- #1

a358ask

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Let ##(X,d)## be a complete metric space and ##g : X \rightarrow X## be a map such that ##\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)## for some ##0<\lambda < 1##.Then ##g## has a unique fixed point ##x^* \in X ##, and it attracts everything, i.e. for any ##x_0 \in X## , the sequence of iterates ##x_0, g(x_0), g(g(x_0))##, ... converges to the fixed point ##x^* \in X##.

The hint I am given are for existence and convergence - prove that the sequence is Cauchy. For uniqueness, choose two fixed points of ##g## and apply the map to both.

My approach is After we got to ##d(x_{n+1}, x_n) \le \lambda^n d(x_1, x_0)##,

then assuming that ##m > n##, ##d(x_m, x_n) \le d(x_m, x_{m-1}) + d(x_{m-1}, x_{m-2}) + ... + d(x_{n+1}, x_{n})## since each term of the right hand size is 0 ...so if we add up all the 0 terms, we get 0 on the right hand size. Therefore, ##d(x_m, x_n)## is 0. If this is right, then the question I have here is how do I guarantee that ##x_{m-1} > x_n##?