A perhaps silly question about > 0

Now and then I see domains defined as in this example:

1) $\displaystyle x \geq \epsilon > 0$

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) $\displaystyle x > 0$

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

The geometric series might be a good example

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

$\displaystyle \sum_{n=1}^\infty z^n$

z being a complex number. It is known that said series is convergent within $\displaystyle |z| < 1$.

However, for claiming in addition that the convergence is uniform it is probably more correct to define a disk of uniform convergence as

$\displaystyle |z| \leq 1-\epsilon < 1$

for any $\displaystyle \epsilon > 0$, no matter how small.

Exactly as you suggested, in such case $\displaystyle |z| < 1$ 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 $\displaystyle |z|$ closer and closer to 1. Whereas chosing a fixed value for $\displaystyle \epsilon > 0$ would allow us to find an index $\displaystyle N_{\delta}$, function only of a $\displaystyle \delta>0$, at which we could stop adding the terms $\displaystyle z^n$ so that

$\displaystyle \left|\sum_{n=1}^{N_{\delta}} z^n -\frac{1}{1-z}\right|<\delta$

is valid for any $\displaystyle z$ in the domain $\displaystyle |z| \leq 1-\epsilon < 1$

Thanks for the tip.

Luca