For my analysis homework I am asked to prove or disprove that if you switch the quantifiers of the definition of a limit, then the definitions are equivalent. More specifically we are asked to prove or disprove for $\lim_{x\to c}f(x)=L$ that these definitions are equivalent,
$\forall \epsilon>0\; \exists\delta>0\; s.t. \forall x\in A\; \text{if}\; |x-c|<\delta, \text{then}\; |f(x)-L|<\epsilon $
$\exists\delta>0\; \forall \epsilon>0 s.t. \forall x\in A\; \text{if}\; |x-c|<\delta, \text{then}\; |f(x)-L|<\epsilon $
My idea is that they are not equivalent. My reasoning is that in the second definition there is an existence claim for delta followed by for all epsilon. This would imply that epsilon depends on delta which would make every limit exist because there is a delta for every epsilon meaning $|x-c|<\delta$ would always imply $|f(x)-L|<\epsilon$.
I am wondering if this is correct reasoning or if I am missing something. Thank you!
Best Answer
I cannot agree or disagree with you, since I fail to see why is it that you say that “This would imply that epsilon depends on delta”.
Anyway, the second assertion implies that $f(x)$ is constant and equal to $L$ on $(c-\delta,c+\delta)$ (since, on that interval, $|f(x)-L|$ is smaller than every number greater than $0$), and therefore, if, say, $f(x)=x$, then the first assertion holds (with $c=L=0$), but not the second one.