The stronger of two propositions
Task number: 2785
Which of the following two propositions is stronger?
- \(\forall x\ \exists K>0: |f(x+1)-f(x)|\le K\)
- \(\exists K>0\ \forall x: |f(x+1)-f(x)|\le K\)
(We say that proposition A is stronger than B if, given that B is true, we may deduce that A is true.)
Hint
In the first proposition the choice of \(K\) may depend on \(x\); in the second there is no such dependence.
If the second proposition is true, we know the value of \(K\) and we can use it for any \(x\) in the first proposition.
This approach does not work in the other direction, since it may be that for various values of \(x\) we must choose various values of \(K\).
For the function \(f(x)=x^2\) the first proposition is true by choosing \(K=2x^2+2x+1\). The second is not true; for a given \(K\) we may choose \(x=\frac{K}2\).
Resolution
The second proposition is stronger.
