Thread: A perhaps silly question about > 0

Now and then I see domains defined as in this example:
1)
for any epsilon greater than zero, no matter how small.

I would be tempted to believe that the above cannot be much different from
2)
as in my understanding that already excludes the case x=0.
Yet, there must be some subtle reason why some author prefers to explicitly write it as in 1)

Can it be just a matter of conventions ? or am I missing some fundamental definition ?
Thanks
Luca

Good question. Obviously, this will depend on the actual issue, but I can give an example of why that might happen.

Very often, a function will converge uniformly everywhere except in a neighbourhood of some point, say 0, where perhaps the function is undefined and around which it does not behave well. Then we cannot say it converges uniformly for x>0, but we can say that it converges uniformly for .

Thanks Bruno,
I believe I have finally got it.
And perhaps, the geometric series might be a good example.

Let us consider the geometric series



z being a complex number. It is known that said series is convergent within .
However, for claiming in addition that the convergence is uniform it is probably more correct to define a disk of uniform convergence as



for any , no matter how small.

Exactly as you suggested, in such case would not be sufficient for uniform convergence, as it would result in the need to add more and more terms to get sufficient accuracy as we choose closer and closer to 1. Whereas chosing a fixed value for would allow us to find an index , function only of a , at which we could stop adding the terms so that



is valid for any in the domain

Thanks for the tip.

Luca