Why order matters in the limit definition

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Hi CStudent, and welcome to MHB!

The condition $$*\ \ \forall N\in\Bbb{N}\ \exists\epsilon>0 \text{ s.t. } \forall n>N \text{ we get } |a_n-L|<\epsilon$$ does not imply that $$\lim_{n\to\infty}a_n = L.$$

For an example of how this can go wrong, suppose that $a_n = 1$ for all $n$ and that $L=0$. If you choose $\epsilon = 2$, the condition $|a_n-L|<\epsilon$ becomes $|1-0|<2$. That condition certainly holds for all $n$, but it is not true that $$\lim_{n\to\infty}a_n = L.$$ In fact, $$\lim_{n\to\infty}a_n = 1$$, which is different from $L$.

That example illustrates that in definitions like this it is very important to put things in the correct order. In the definition of limit, the choice of $\epsilon$ comes first, and then the choice of $N$ (which will usually depend on $\epsilon$) comes next. But in the condition $*$, you have put the $N$ before the $\epsilon$. That changes the whole meaning of the statement.

Your other condition $*$ is also not equivalent to the definition of limit.

But your final condition (the statement that $$b = |a_n-L| \to0\ \Longrightarrow\ \lim_{n\to\infty}a_n = L$$) is correct.
 
Opalg said:
Hi CStudent, and welcome to MHB!

The condition $$*\ \ \forall N\in\Bbb{N}\ \exists\epsilon>0 \text{ s.t. } \forall n>N \text{ we get } |a_n-L|<\epsilon$$ does not imply that $$\lim_{n\to\infty}a_n = L.$$

For an example of how this can go wrong, suppose that $a_n = 1$ for all $n$ and that $L=0$. If you choose $\epsilon = 2$, the condition $|a_n-L|<\epsilon$ becomes $|1-0|<2$. That condition certainly holds for all $n$, but it is not true that $$\lim_{n\to\infty}a_n = L.$$ In fact, $$\lim_{n\to\infty}a_n = 1$$, which is different from $L$.

That example illustrates that in definitions like this it is very important to put things in the correct order. In the definition of limit, the choice of $\epsilon$ comes first, and then the choice of $N$ (which will usually depend on $\epsilon$) comes next. But in the condition $*$, you have put the $N$ before the $\epsilon$. That changes the whole meaning of the statement.

Your other condition $*$ is also not equivalent to the definition of limit.

But your final condition (the statement that $$b = |a_n-L| \to0\ \Longrightarrow\ \lim_{n\to\infty}a_n = L$$) is correct.

Great, thank you1