Question
Consider the set S of rational numbers discussed prior to thestatement of the completeness Axiom, as well as the numbers p and qdefined there. Prove each of the following. Let S be a nonempty set of real numbers. a)The set S is bounded above if there is a number M such thatx M for all x in S. The number M is called an upperbound of S. b)The set S is bounded below if there is a number m such thatx m for all x in S. The number m is called a lowerbound of S. Completeness Axiom: Each nonempty setof real numbers that is bounded above has a supremum. a)If y is a rational number such that y2 > 2,then y is an upper bound of S. b)Every rational number that is an upper bound of S is greaterthan 1. c)The number q is rational.
Explanation / Answer
for all the bits (a) , (b) , (c) the answer is the set ofrationals is not complete. because , if we consider the set S of rational numbersless than 2 , then any rational number > 2 is an upper bound ofS and so , S is bounded above, but if we assume tobe theleast upper bound of S , then both > 2 and < 2are possible which is an absurdity. so, it follows that S is not complete. i.e. any subset of rationals is not complete,. so, Q is not complete.