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TIFR-2017-Maths-B: 5

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Prove or disprove the following: let $f:X\rightarrow X$ be a continuous function from a complete metric space $\left ( X,d \right )$ into itself such that $d\left ( f\left ( x \right ),f\left ( y \right ) \right )< d\left ( x,y \right )$ whenever $x\neq y$. Then $f$ has a fixed point.
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