A bit of overcomplicating on interview question
Question is the following: Solve x^x^x^... = 2.
So, because of this condition we have x^2=2, and from that we can conclude that only possible solution is x=sqrt(2). But how do we know that this is actually a solution, we know that x=sqrt(2) is only candidate for solution, but haven't proved that it is actually solution.
It seems to me intuitively that left-hand sidr of equation is a continuous function, and then the result would be obvious from BW theorem, but I am not 100% sure that we are talking about continuous function.
Sorry for bothering you with this, but I would really like to be 100% sure about the answer.
Don't think of it as a continuous function, think of it as an exponential series. You're already given from the original equation that your infinite function converges to a finite integer. You don't need to prove that fact because it is a given.
Think of this backwards via n notation: For any n, x(n) = x^[x(n-1)] so for example x(3) = x^(x^x) = x^x(2) So for a converging series, you can eventually see that x^y = y (where y exists and is finite). Given that y is 2 in your original equation, you know the thing to solve is x^2 = 2, and you have x = sqrt(2).
It's been ages since real analysis, I don't remember any of it, not even BW thereom. Do remember HB, but fuck if I remember how to apply it.
I also doubt your interviewers care about the real analysis-based proofs behind your solution. Simply showing them through common sense that 2 = x^(x^x^x^x.... = 2) should be enough.
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