Make a picture of the usual spiral converging to the fixed point, in the style of 
Where does this little obsession aparent on this site with 'layman' explanations? :P
NB. The picture comes from the Wikipedia page on fixed points.
Later. To make the picture it is better to use the cosine (so you do not even have to make the picture because that is what's in the wikipedia page) mainly because the iteration is seen more clearly: for the square root, the sequence converges boringly and fast to be interesting.
PS. You can draw this kind of pictures, assuming you have acces to Mathematica, with the following code:
f[t_] := Cos[t];
g = Plot[{f[t], t}, {t, -\[Pi]/2, \[Pi]/2},
AspectRatio -> 1, PlotStyle -> {Thick}, Epilog ->
Module[{pts = NestList[f, .1, 10]},
{
Thin, PointSize[0.015],
Line@ Flatten[Map[{#, {#[[2]], #[[2]]}} &, Partition[pts, 2, 1], 1], 1],
Red, Point[Partition[pts, 2, 1]]
}
]
]