The claim about starting points is baffling (and certainly not carefully tested). Try printing the starting point at each iteration. You will notice below that starting points in each iteration come from immediately prior results.

In[74]:= FoldList[(Print[#1]; 
   x /. FindRoot[x^2 - #2*x + 1, {x, #1}]) &, .5, Range[3, 6]]

During evaluation of In[74]:= 0.5

During evaluation of In[74]:= 0.38196601125

During evaluation of In[74]:= 0.267949192431

During evaluation of In[74]:= 0.208712152522

Out[74]= {0.5, 0.38196601125, 0.267949192431, 0.208712152522, \
0.171572875254}

It's a different set of equations from before. In this case each one has .5 as solution and each has .5 as starting point. Since solutions and starting points are all the same, I have no idea what conclusion might be drawn from this in regards to whether the starting point is selected from prior solutions or from the initial starting point.

FoldList can be useful for this sort of thing.

FoldList[x /. FindRoot[x^2 - #2*x + 1, {x, #1}] &, .5, 
 Range[3, 6]]

(* Out[76]= {0.5, 0.38196601125, 0.267949192431, 0.208712152522, \
0.171572875254} *)