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58 The Nuts and Bolts of Proof, Third Edition
4. Let (2 > 1 be a fixed number. Then for all integer numbers fc > 3,
5. Check that for all counting numbers fe > 1,
6. The inequality k^ < 5/c! is true for all counting numbers /c > 3.
7. Let n be an odd counting number. Then n^ — 1 is divisible by 4.
8. The number 10^ - 1 is divisible by 9 for all /c> L
9. Let n be an integer larger than 4. The next to the last digit from the
right of 3" is even.
10. Let ^ = ( { J Y Then ^" = ( ^ J ) for all n>2. (*)
EXISTENCE THEOREMS
Existence theorems are easy to recognize because they claim that at least one
object having certain properties exists. Usually this kind of theorem is
proved in one of two ways.
1. If it is possible, use an algorithm (a procedure) to construct explicitly
at least one object with the required properties.
2. Sometimes, especially in more advanced mathematical theories,
the explicit construction is not possible; therefore, we must be
able to find a general argument that guarantees the existence of the
object under consideration, without being able to provide an actual
example of it.
EXAMPLE 1. Given any two distinct rational numbers, there exists a third
distinct rational number between them.
Discussion: We will reword the statement above:
A. Consider two distinct rational numbers, a and fc, with a> b.
Implicit hypothesis: Because the two numbers are not equal, we can
assume without loss of generahty that one is smaller than the other.
We can use all the properties and operation of rational numbers.