Given A, B, and C are members of the set of real numbers. Prove indirectly that, if A<B and B<C, then A<C?

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Answer 1 · 2017-10-30T21:33:32

See explanation...

If #RR_+ sub RR# is the set of positive real numbers, then we can define:

#A < B" " <=> " " EE d in RR_+ : B = A+d#

Then given:

#{ (A < B), (B < C) :}#

we can deduce that #EE d_1, d_2 in RR_+# :

#{ (B = A+d_1), (C = B+d_2) :}#

So:

#C = B+d_2 = A+d_1+d_2 = A+d#

where #d = d_1+d_2 in RR_+#

Hence:

#A < C#