I found this false lemma in a math text online:

Suppose f: A --> B is a monotone bijection between two subsets of R (the real numbers).  Then f is a homeomorphism

 

whew!  Somebody needs some coffee!

This was in a text that was written by a math professor for an undergraduate course.

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However, the lemma is true if you add the condition that A and B are compact (equivalent to closed and bounded for R). 

I wish I had known someone like you when I was taking analysis at Penn State. Maybe I would have survived.

why thanks.  This text was used in a course at Penn State actually.  It's an introduction to p-adic analysis that was used in an undergraduate class.  It's very good in some ways - very good exercises, well organized, etc.  But I was facepalming over this flub in the text - it seems amazing that a math professor could believe the false lemma was true, even for a minute.  I've been adding comments to the text as I go through it. 

I was always curious about p-adic analysis - p-adics are an alternative metric completion for the rational numbers.  Now I'm learning the basics of it. 

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