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Theorem 1:

THEOREM 1

Let a, b and c be hyperreal numbers.

(i) a ≈ a.

(ii) If a ≈ b, then b ≈ a.

(iii) If a ≈ b and b ≈ c, then a ≈ c.

These properties are useful when we wish to show that two numbers are infinitely close to each other.

The reason for (i) is that a - a is an infinitesimal, namely zero. For (ii), we note that if a - b is an infinitesimal ε, then b - a = -ε, which is also infinitesimal. Finally, (iii) is true because a - c is the sum of two infinitesimals, namely a - b and b - c.


Last Update: 2006-11-05