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I found this exercise:

$f:K \rightarrow K$ where $K$ is compact and $||f(x) - f(y)|| = ||x - y||$, for all $x, y$ in K, implies $f$ is bijective. Hint: show that $f$ is injective and continuous and consider the sequence $x_j = f(x_{j−1})$, where $x_0 \notin f(K)$. Show that $||x_j - x_{j+m}|| = ||x_0 - x_m||$ and arrive at a contradiction.

I've been able to show that $f$ is injective and continuous, but I have no idea how to use the hint to show surjection. Any ideas? Thanks in advance.

Arthur
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    Welcome to MSE. Here's how to ask a good question. Follow these guidelines to get help in this forum. For example, there you'll find the following: "Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title." – jjagmath Feb 23 '24 at 01:33
  • @jjagmath Thanks for pointing it out. I hope it's better now. – Arthur Feb 23 '24 at 01:48
  • HINT: What do you know about sequences in a compact set and what do you know about distance from a compact set to a point in its complement? – Ted Shifrin Feb 23 '24 at 02:03
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    @Arthur It's great that you're willing to edit your post to better align with our policies, so please don't take it personally when I say, to many people here, you have not provided enough context to your question. I would add your proof that $f$ is continuous and surjective, and if you had any thoughts/aborted attempts at proving surjectivity, I would include them too, up to where you got stuck. So far, your question has two close votes and one down vote, which is to say, at least two other people seem to think your question needs more. :) – Theo Bendit Feb 23 '24 at 02:14
  • You don't need compactness to prove injective and continuousl so you probably need to use that somewhere. – Thomas Andrews Feb 23 '24 at 02:22
  • Presumably, $K$ isn't just a compact space, since $x-y$ and $f(x)-f(y)$ are not defined in any old compact space. Presumably, $K$ means some compact subspace of a horned vector space? – Thomas Andrews Feb 23 '24 at 02:23

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