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**tttcomrader** Consider the equation $\displaystyle x^2-x-c=0, \ \ \ x>0 $

Define the sequence $\displaystyle \{ x_n \} $ by $\displaystyle x_{n+1} = \sqrt {c+x_n} \ \ \ x_1 >0 $

So that the sequence converges monotonically to the solution of the equation.

Proof so far.

I have $\displaystyle x_2 = \sqrt {c +x_1 } $ and $\displaystyle x_3 = \sqrt {c+x_2} = \sqrt {c + \sqrt {c+x_1 } } > \sqrt {c+x_1} = x_2$

So this sequence is monotonically increasing, but do I need to do induction to be more convincing?