It’s one of my real analysis professor’s favourite sayings that “being obvious does not imply that it’s true”.

Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).

But what are some theorems (preferably short ones) which, when put into layman’s terms, the average person would claim to be true, but, which, actually, are false

(i.e. counter-intuitively-false theorems)?The only ones that spring to my mind are the Monty Hall problem and the divergence of ∞∑n=11n (counter-intuitive for me, at least, since 1n→0

).I suppose, also, that lim is not obvious, since one ‘expects’ that \left(1+\frac{1}{n}\right)^n \to (1+0)^n=1.

I’m looking

justfor theorems and not their (dis)proof — I’m happy to research that myself.Thanks!

**Answer**

**Theorem** (false):

One can arbitrarily rearrange the terms in a convergent series without changing its value.

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