I have to make the negation of the following sentence.
There is some $\epsilon>0$ such that $f(x)>\epsilon~$ for all $x>0$.
Here is my attempt:
For each $\epsilon>0$, $f(x)\leq \epsilon~$ for some $x>0$.
Am I correct?
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Correct. The initial statement reads $$\exists \varepsilon >0,\ \forall x>0,\ f(x)>0. $$ When negated, we switch the quantifiers and negate the statement $f(x)>0$. So $$ \forall \varepsilon >0,\ \exists x>0,\ f(x)\leqslant 0. $$
Note that the statment does not explicitly make use of $\varepsilon$ anywhere, though. Could replace $f(x)>0$ with $f(x)\geqslant\varepsilon$, for instance.
AlvinL
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$$\text{There is some $\epsilon>0$ such that ($f(x)>0~$ for all $x>0$)}$$
$$\text{(There is some $\epsilon>0$ such that $f(x)>0~$) for all $x>0$}$$ – GEdgar Feb 05 '23 at 11:10