A proposition with functions
Task number: 2784
Does the function \(f(x)=\sin x\) satisfy the following proposition, or its negation? \[ \forall \varepsilon>0\ \exists K>0\ \forall x: x>K \Rightarrow |f(x)|<\varepsilon.\]
Resolution
The negation of the proposition reads: \( \exists \varepsilon>0\ \forall K>0\ \exists x: x>K \land |f(x)|\ge \varepsilon\).
This negated proposition is satisfied by choosing \(\varepsilon=\frac12\) and \(x=\frac\pi2+\lceil K\rceil\pi\).
Result
The proposition is not true for the given function.
