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.

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
Cs translation
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