Let $ f : X \to Y $ a biyective funtion and uniformly continuous.

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Let $ f : X \to Y $ a biyective funtion and Uniformly continuous.Prove that is Y is complete and $f^{-1}$ is continuous, then X is complete.

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Hints:

(1) Take a Cauchy sequence $(x_n)_{n\in\Bbb N}$ in X.

(2) $(f(x_n))_{n\in\Bbb N}$ will be Cauchy (why?).

(3) $(f(x_n))_{n\in\Bbb N}$ will be convergent (why?).

(4) $(x_n)_{n\in\Bbb N}$ will be convergent (why?).