I Why Must Epsilon Be Greater Than Zero in Sequence Limits?

[Shlomi93]

in the limit definition of a sequence, why does epsilon has to be greater than 0 and not greater or equal to 0?

thanks in advance.

 

[fresh_42]

in the limit definition of a sequence, why does epsilon has to be greater than 0 and not greater or equal to 0?

thanks in advance.

It doesn't really matter with sequences of real numbers. You could take both as you can always find another epsilon that is slightly smaller. In general, however, one speaks of open neighborhoods around the limit point as they are the defining element of general (topological) spaces. And open translates to smaller than, whereas smaller or equal includes the boundaries, and as such are closed sets. So the restriction to smaller than is somehow simply consequent, even if not needed (and it's available on the keyboard).

 

[pwsnafu]

Choose $\epsilon = 0$, then in order for $a_n \to L$ we need to find a $N$ such that $a_n = L$ for all $n>N$.
So under this definition the only sequences that converge are those that are eventually constant.

 

[fresh_42]

Choose $\epsilon = 0$, then in order for $a_n \to L$ we need to find a $N$ such that $a_n = L$ for all $n>N$.
So under this definition the only sequences that converge are those that are eventually constant.

Yes, you're right. I confused it with the condition $\,\vert \,a_n -L\,\vert \, < \varepsilon$ where you could take $\leq$ instead.
Of course $\varepsilon = 0$ would make no sense as there would be only constant sequences left over.

 

[Shlomi93]

Yes, you're right. I confused it with the condition $\,\vert \,a_n -L\,\vert \, < \varepsilon$ where you could take $\leq$ instead.
Of course $\varepsilon = 0$ would make no sense as there would be only constant sequences left over.

Choose $\epsilon = 0$, then in order for $a_n \to L$ we need to find a $N$ such that $a_n = L$ for all $n>N$.
So under this definition the only sequences that converge are those that are eventually constant.

thank to both of you!

 

[Mark44]

in the limit definition of a sequence, why does epsilon has to be greater than 0 and not greater or equal to 0?

Consider $a_n = \frac {n - 1} n, n \ge 1$. It's easy to show that $\lim_{n \to \infty}a_n = 1$. However, if $\epsilon = 0$, it's not possible to find a specific number N for which $|a_n - 1| = 0$, for all $n \ge N$.

With $\epsilon > 0$, all that has to happen is to force the terms in the tail of the sequence arbitrarily close to L, not necessarily making them exactly equal to it.