Square root, expressed as the limit of a sequence
Let \( c \in \mathbb{R} \), \( c > 0 \), be a positive real. The sequence defined recursively below converges, and it converges to \( \sqrt{c} \).
Let \( x_1 = c \), and let \( x_n \) be defined like so:
Proof by induction (any other typical methods) can be used to show that sequences defined recursively like this do (or don't) converge.
Intuition
The outer \( \frac{1}{2} \) factor can be seen as taking an average of the two inner terms. Inspecting the second term, \( \frac{1}{2}(c + \frac{c}{c}) = \frac{c + 1}{2} \), the third, \( \frac{1}{2}(\frac{c+1}{2} + c \frac{2}{c+1}) \), and so on, one can be convinced that this sequenc is decreasing starting from the third term. In addition, it can be seen that the terms don't fall below \( \sqrt{c} \). While not exactly convincing as a proof, you can check if a term did surpass \( \sqrt{c} \), then the following term would be less than that term.