Explanation: A circle of unit area can be sliced up in any way and all of its parts added together will always equal 1. This circle is sliced up into ever-thinner pie slices where the first slice leaves 90% untouched (0.9), the second slice leaves 90% of the remaining wedge untouched (0.09), the third slice leaves 90% of that remaining wedge untouched (0.009), and this process continues without end. The summation of all slices produces the infinite series:

0.9 + 0.09 + 0.009 + 0.0009 + … = 0.9999…

And because the slicing never changes the total area of this circle with an area of 1 (both prior to the slicing and after), while the amount of circle not included in the sum goes to 0, the conclusion is that
0.9999… = 1.

Mathematica code that produced this image (before hand editing):

diagramBlue = RGBColor[0.4, 0.8, 1];
coord[n_Integer] := {
  Cos[(1/4 + 10^(-n)) 2 \[Pi]],
  Sin[(1/4 + 10^(-n)) 2 \[Pi]]
  }
radius[n_Integer] := Line[{{0, 0}, coord[n]}]
pointLabel[label_String, points_List : {x_, y_}] := Text[
  TextCell[
   label,
   "Text",
   CellFrame -> True,
   CellFrameMargins -> 0.1,
   CellFrameColor -> diagramBlue,
   Background -> White
   ],
  points
  ]
diagram = Show[
  Graphics[{
    {diagramBlue, EdgeForm[Black], Disk[]},
    radius[0],
    radius[1],
    radius[2],
    radius[3],
    pointLabel["0.9\n36°", coord[1] + {-0.05, 0.11}],
    pointLabel[
     "0.09\n3.6°", {coord[2][[1]] - 0.05, coord[0][[2]] + 0.12}],
    pointLabel[
     "0.009\n0.36°", {coord[3][[1]] + 0.05, coord[0][[2]] + 0.12}]
    }],
  ImageSize -> 500
  ]
Export[FileNameJoin[{Directory[], "Downloads", 
   "0.999... = 1 CircleGraphProof.svg"}], diagram]