I'm asked to find if the fixed-point iteration 

$$x\_{k+1} = g(x\_k)$$

converges for the *fixed points* of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$.

In this short video by Wen Shen,

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> <https://www.youtube.com/watch?v=3_PKDDnxgCo>
> 
> 
> 

it's explained how to find these fixed-points and to see if a fixed-point iteration converges. My doubt is related to find if a fixed point iteration converges for a certain fixed point.

At more or less half of the video, she comes up with the following relation for the error

$$e\_{k+1} = |g'(\alpha)| e\_k$$

where $\alpha \in (x\_k, r)$, by the mean value theorem, and because $g$ is continuous and differentiable.

If $|g'(\alpha)| < 1$, then the fixed-point iteration converges. 

I think I agree with this last statement, but when she tries to see if the fixed point iteration converges for a certain root of a certain function, she simply finds the derivative of that function and plugs in it the root.

I don't understand why this is equivalent to $$e\_{k+1} = |g'(\alpha)| e\_k$$

Can someone fire up some light on my brain? (lol, can I say this?)