Complete bounded metric space


Let (X,d) be a complete bounded metric space and let G be the set of
all homeomorphisms from X onto itself.
Define a metric in G as follows:

D(f,g) = sup { d(f(x),g(x)) + d(f^(-1)(x), g^(-1)(x)) }

Prove that (G,D) is a complete metric space. Could you please give
some hints? I'm stuck.
.

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