Carefully formulate the negation of the following statement.Then prove the state
ID: 2938861 • Letter: C
Question
Carefully formulate the negation of the following statement.Then prove the statement by contradiction. There is no greatest negative real number. Carefully formulate the negation of the following statement.Then prove the statement by contradiction. There is no greatest negative real number.Explanation / Answer
The negation is the following: For every neg real x there exists neg real y such that x > y. To prove this by contradiction we need to first formulate an if - then statement as follows: if x is neg real, then there is a neg real y such that x > y. now this statement has the form if P then Q. Now to execute a contradiction proof assume P and NOT(Q), that is, assume x is a neg real and that there is NO neg real y such that x > y. Now consider the sequence, -1/n, where n takes values in the natural numbers. Then, the first term is -1,second term is -1/2, third term is-1/3, etc. This sequence tends toward 0 but never quite gets there. Now we can assume that x is between -1 and 0 exclusive. Therefore, there exists to terms, k and L, in this sequence such that -1/k>x>-1/L. Now this second inequality is our contradiction x > -1/L. QED