Question

11.1 The Method of Exhaustion

(a) Assuming the so-called axiom of Archimedes: If we are given two magnitudes of the same kind, then we can find a multiple of the smaller that exceeds the larger, establish the basic proposition of the method of exhaustion: If from any magnitude there be subtracted a part not less than its half, from the remainder another part not less than its half, and so on, there will at length remain a magnitude less than any preassigned magnitude of the same kind.

Answer 1

If from any magnitude there be subtracted a part not less than its half, from the remainder another part not less than its half, and so on, there will at length remain a magnitude less than any preassigned magnitude of the same kind

Let's establish above statement by 'Proof by construction' method by taking an example

Let us suppose that we have a length of magnitude 10. Subtract 8(length not less than half) from 10.

10 - 8 = 2

2<10

Now subtract 1.5 from 2

2-1.5 = 0.5

0.5<2

Now subtract 0.3 from 0.5

0.5 - 0.3 = 0.2

0.2<0.5

Now subtract 0.15 from 0.2

0.2 - 0.15 = 0.05

0.05<0.2

Now subtract 0.03 from 0.05

0.05 - 0.03 = 0.02

0.02<0.05

Now subtract 0.016 from 0.02

0.02 - 0.016 = 0.004

0.004<0.02

And so on

We will always get a magnitude less than any preassigned magnitude.