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“Why do I always get 1 when I keep hitting the square root button on my calculator?”

Take any positive number above or below 1. Upon performing repeated square root operations the end result is always 1. Is there a theorem in Math that can prove this rather intriguing result? Thank you. Klim

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marked as duplicate by Aryabhata, Bill Dubuque, Rahul, Hans Lundmark, Qiaochu Yuan Feb 28 '11 at 12:53

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1 Answer 1

You have the sequence $x_{n+1} = \sqrt{x_n}$

For $0 \lt x_1 \lt 1$, we have that $x_{n+1} \gt x_n$ and $x_n \lt 1$. So this is a monotonically increasing sequence bounded above, and so has a limit.

The limit $L = \lim_{n \to \infty} x_n$ satisfies $0 \lt x_1 \le L \le 1$ and $L = \sqrt{L}$, thus $L = 1$.

If $x_1 \gt 1$, you get a monotonically decreasing sequence which is bounded below and you get $\lim_{n \to \infty} x_n = 1$, similarly.

Note that, if you don't start with $1$ (or $0$), in a finite number of steps, you will get very close to $1$, but don't really attain $1$. The result of $1$ you see is likely due to the precision limitations of your calculator.

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