Contracting a metric space

The following problem comes from this nice book (and I suppose also from other sources).
As anybody knows a contraction on a metric space has a fixed point.
Now suppose is a metric space and is not exactly a contraction but satisfy
Suppose also that X is compact, then a fixed point must exists for and for any .

Here is a hint

Here is a solution sketch

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Hi guys, I’m Marco and this is a blog devoted to collect and discuss the best undergraduate level math problems just for entertainment. If you know any funny or tricky exercise to suggest send me a message and I will make a post about it.