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I need hints to prove the implication “path-connected implies arc-connected” for metric spaces? [from Pugh chapter $2$ exercise $75$]

For simplicity, let me just call, given a path $f : [0,1] \to M$, a pair of points $(a,b)$ such that $f(a)=f(b)$ for distinct $a$ and $b$ an intersection point.

This exercise problem seems trivial enough if the number of intersection points is only finite. We can just remove all the inbetween image of intersection points $f( (a,b) )$ and get a homeomorphism. However, I cannot seem to tackle the case where there are an infinite number of intersection points.

I tried doing stuff like considering the supremum and infimum of the set of intersection points but that led nowhere because of some strange cases where the path is just very ugly/not convenient.

Pugh says this is only $1$ star difficulty which is why I feel like I’m just missing something. A hint in the right direction ( or even a complete proof ) would be very much appreciated. Thanks

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    $\begingroup$ Welcome to MSE! In the future, try to type math with MathJax (pretty much the same as LaTeX if you're familiar with that one). $\endgroup$
    – Bruno B
    Commented Jun 26, 2023 at 10:06
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    $\begingroup$ IMHO, this is an extremely difficult exercise. I do not see that the metric case should be considerably easier than the case of general Hausdorf topological spaces. $\endgroup$
    – Jochen
    Commented Jun 26, 2023 at 12:16
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    $\begingroup$ Hi, please stop flagging my question because this is no duplicate. I try to be kind while typing, but, frankly, this is ridiculous. Anybody who has any knowledge at all regarding basic undergraduate calculus knows that arc-connectedness is a completely different concept, definitionally, to connectedness. Is the person flagging my question a high school kid or someone who has not enough respect to spare me to even bother reading my question. This is the second time I’ve been flagged and now my post is taken down. $\endgroup$
    – Fernandeez nuts
    Commented Jun 29, 2023 at 17:45
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    $\begingroup$ @Fernandeeznuts: This is a very common occurrence, unfortunately. Most users here do not even bother to read questions when choosing to mark them as duplicates. $\endgroup$
    – Eric Wofsey
    Commented Jun 29, 2023 at 18:09
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    $\begingroup$ @TheSilverDoe I have attempted to look at answers to this same question but in terms of Hausdorff spaces before; however, it is way beyond my realm of understanding. My question, as written, came from Pugh ( a very undergraduate-level analysis book) My question actually mostly comes from a place of trying to understand an undergraduate-version proof of this theorem ( without using complicated topology or measure theory concepts ). I raised this question because, as stated also, the question was rated 1 star (meaning average difficulty) by the author. $\endgroup$
    – Fernandeez nuts
    Commented Jun 29, 2023 at 18:43

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