Formal definition of limits

1/n =< 1/N =< (E + 1)

1/n - 1 =< 1/N - 1 =< E

Here is a flaw.

If I have a sequence $a_n$, and it converges to a limit $a$ then

$\forall \varepsilon >0 ,\varepsilon \in \mathbb{R} \quad \exists N(\varepsilon) \in \mathbb{N} : \forall n \geq N(\varepsilon) : |a_{n}-a| < \varepsilon$

Note the absolute values.

Now you

have that if $1/n \leq 1/N \leq (E + 1)$ then $(1/n) -1 \leq (1/N) -1 \leq E$

We've just subtracted 1 so its still ok.

But it does not then follow that $\mid (1/n) -1 \mid< E$

For example $-8 < 1$ but $\mid-8\mid >1$

So you cannot conclude the proof.

Thank you very much for replying! But I'm still a little confused. How come we have no problem in the first example?

I said there that 1/n =< 1/N =< E

And then I said that

|1/N| < E

How come it follows here that |1/N| < E ??

I know that in this example it is obvious that the limit is 0 and I can see now why my alternative proof was wrong. But how do I know that the first proof is correct. It seems to me that I am making the assumption that the limit is 0 from the start.

Like, how would you have approached this proof if you didn't know the limit beforehand? How would you decide what N is?

And (sorry for all the questions) how is the first proof proving anything????

The reason why it follows is that $1/n$ is always positive. You had $1/n < E$. It then follows that $\mid 1/n \mid <E$ as they are both positive.

But in the second example, it is not always true that $(1/n) - 1$ is positive. Remember in my example, it was the minus that meant we could not preserve the inequality.




Well let us note a few things to clear some confusion:

1 - This method of proving the convergence of a sequence needs a candidate for the limit. So to find a candidate for the limit we use our intuition and (sometimes) some unrigorous mathematics.

2 - Just take a moment to look at what the definition says without the epsilons and Ns. It says that if I have a sequence that converges to a limit, then I can go far enough down the sequence (or take enough terms, whichever you prefer) such that the distance between my limit and the sequence is some fixed positive number (we are only interested in small epsilons for the sake of convergence)

3 - If I didn't know the limit beforehand, well that depends. If I could work out the limit using intuition or some estimation, then you could attempt to prove it like this. Otherwise, there are other methods of proof. For example, Cauchy's General Principle of Convergence can be used to show the convergence of a sequence without finding the limit.

4 - Bear in mind too, it is not necessary to find N. The proof does not require it. All that is required is for you to demonstrate the existence of 1 such N that works

Thank you so much, this has cleared everything up for me!!!