Here is a problem from fakalin. Find the error in the following proof:

Let \(x^{x^{x\cdots}} = 2\). Then \(x^{x^{x\cdots}} = x^{x^{x^{x\cdots}}} = 2\). Substituting, we get \(x^2=2\) and \(x=\sqrt{2}\). Now let \(x^{x^{x\cdots}} = 4\). Similarly, \(x^{x^{x^{x\cdots}}} = 4\), \(x^4=4\) and \(x=\sqrt{2}\). But then \(2=x^{x^{x\cdots}}=4\) and \(2=4\).
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According to this distribution, the mild amount of hyperopia (far-sightedness) I have in both eyes is in statistically acceptable ranges, and is easily corrected optically. Even without correction, this amount of hyperopia — unlike myopia — is such that I could and did get away with normal accommodation. After a visit to a very thorough ophthalmologist, though, I found out that refractive error was the least interesting thing about my vision, and as a result I finally have answers to everything that I had wondered about for years. For example, now I know why it was more difficult for me to see those Magic Eye images.
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