I am sure many of you have spotted the mathematical nonsense of my coding above - as it occurred to me in a sleepless night. What is wanted is of course, the numbers of a single sequence (here e.g., FareySequence[70]) plotted according to their values and their denominators:

maxVal = 70;
fareyPoints = {#, Denominator[#]} & /@ FareySequence[maxVal];
Plot[{1/x, 1/(1 - x)}, {x, 0, 1}, PlotStyle -> Red, PlotRange -> {0, maxVal}, Epilog -> Point[fareyPoints], ImageSize -> Large]

Now basically, we have a short one-liner!

@chiao-yin hsu: The problem in your code is that you are plotting 70 sequences but without deleting the duplicates - as I did in my first approach (by using the complements successively).

Hello Chiao-Yin Hsu,

according to the reference you cited, this is probably the graphics you are asking for:

maxVal = 70;
fareyData0 = Table[FareySequence[n], {n, 1, maxVal}];
fdp2 = Reverse /@ Partition[fareyData0, 2, 1];
fareyData1 = Complement @@@ fdp2;
fareyData = MapIndexed[Thread[{#1, #2} &[#1, First[#2]]] &, Prepend[fareyData1, {0, 1}]];
fp = ListPlot[fareyData, PlotStyle -> Black];
pl = Plot[{1/x, 1/(1 - x)}, {x, 0, 1}, PlotRange -> {0, maxVal}, PlotStyle -> Red];
Show[pl, fp, ImageSize -> Large]

ADDENDUM: If you just want to see the points you can simply do it like so:

NumberLinePlot[fareyData1, PlotStyle -> Black, ImageSize -> Large]