Let F be increasing on [a.b].
(Definition of increasing: if x \(\le\) y, then f(x) \(\le\) f(y))
If f(x) < f(y), then x < y.
By Contrapositive: If x \(\ge\) y, then f(x) \(\ge\) f(y).
The contrapositive matches the definition of an increasing function(with the variables switched), thus proving that if f(x) < f(y), then x < y.
Looks good! Just a couple of tweaks to perfect it:
ReplyDelete(1) In the title, \( [a.b] \) should be \( [a,b] \).
(2) In the first line, "Let F be..." should be "Let \(f\) be..."
(3) Since proofs should be made up of complete sentences, rather than the parenthesized definition of an increasing function, use something like (with no parentheses) "By definition, if \(x \le y\), then \(f(x) \le f(y) \)..."
(4) Similarly, replace "By contrapositive: ..." with something like "Then by the contrapositive, this implies..."
PS Also please tag your proof with labels relevant to the method (contrapositive) and topics (properties of functions). Thanks!
ReplyDelete