## The stronger of two propositions

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.