# Confusion regarding the definition of limit

We know that, $$\displaystyle\lim_{n\to\infty}x_n=x$$is equivalent to the following statement, $$(\forall \varepsilon>0)(\exists n_0\in\mathbb{N})(\forall n\ge n_0)(\left\lvert x_n-x\right\rvert<\varepsilon)$$

But my question is that if $\displaystyle\lim_{n\to\infty}x_n=x$ is interpreted informally we can say that the limit of $x_n$ as $n$ tends to $\infty$ is $x$ and this is equivalent to saying, $$n\to\infty\implies x_n\to x$$From this perspective, it seems that we need to define the symbols $n\to\infty$ and $x_n\to x$ separately but one of my friend said that it is redundant, and it is of course evident from the standard definition. But I think that it is necessary from this viewpoint. Where am I wrong?

Added:-

I think that the wording of the original question has created some confusion about the focus of it. So, I will try to elaborate this point a bit. Thinking that a large part of Mathematics can be phrased to be "formalization of intuition" I asked the question.

We are told that informally $\displaystyle\lim_{n\to\infty}x_n=x$ means that,

$x_n$ tends to $x$ as $n$ goes to $\infty$.

So, I thought that it is natural to formalize this intuitive statement.

As a first step, we note that the above written statement is equivalent to the following statement,

$x_n$ tends to $x$ as $n$ tends to $\infty$.

The second step is,

$n$ tends to $\infty$ implies $x_n$ tends to $x$.

So, I thought that if we can formalize the word "tends to" we will be finished. For this I took help of the symbol $\to$ and formalized the above statement as,

$n\to \infty \implies x_n \to x$

But this said formalization will be nonsense if we can't express the meaning of $n\to\infty$ and $x_n\to\infty$ formally and it can be done easily by this. So, the conclusion of all these was that, $$\displaystyle\lim_{n\to\infty}x_n=x\iff (n\to \infty \implies x_n \to x)$$but in the same page (the link of which is provided above) we find that, $$\displaystyle\lim_{n\to\infty}x_n=x\iff x_n \to x$$I insist that my confusion is not at all regarding the symbols but their logical (or formal) interpretation which apparently leads to the nonsensical equivalence, $$(n\to \infty \implies x_n \to x)\iff x_n \to x$$

In Mauro Allegranza's answer it is said that 'nowhere we have the concept of "something approaching to something other" '. True. Note that, we also did't need this concept in appointing the symbol $\to$ while formalizing our statement.

• Indeed you are not defining $n \to \infty$ or $x_n\to x$. You are defining "$n \to \infty \Rightarrow x_n \to x$", and the definition is the one you give. – user99914 Mar 29 '15 at 9:14
• @John: But I thought (please correct me if I am wrong) that to be able to "define" a statement $P\implies Q$, the statements $P$ and $Q$ should both be well-defined. – user170039 Mar 29 '15 at 9:18
• Might be I shouldn't use your notation. Think it this way, when you say $x_n \to x$, it is a bit vague: what is actually moving such that $x_n$ moves to $x$? So $x_n \to x$ alone don't make a lot of sense. You have to state something like "$x_n \to x$ as $n \to \infty$". – user99914 Mar 29 '15 at 9:21
• @John: I knew that this answer would come. But what about if we say that $\displaystyle\lim_{n\to\infty}x_n=x$ is logically equivalent to $x_n\to x$ then it doesn't remain vague at all. – user170039 Mar 29 '15 at 9:24
• Then $n \to \infty \Rightarrow x_n \to x$ is not the same thing as $x_n \to x$ (whatever the definition of $n \to \infty$ is). – user99914 Mar 29 '15 at 9:25

## 2 Answers

We are not defining two concepts, but only one :

the limit of a sequence.

A sequence of real numbers : $(x_n)$ is a function :

$f : \mathbb N \to \mathbb R$

i.e. $f(n)=x_n$.

The Limit of the sequence $(x_n)$, if it exists, is a number $L$ such that :

For each real number $\epsilon > 0$, there exists a natural number $N$ such that, for every natural number $n \ge N$, we have $|x_n - L| < \epsilon$.

Thus, we have two "objects" : sequence and limit of a sequence, where the first one is defined independently from the second one.

This is the "formal" translation of the intuitive concept of "a sequence of numbers approaching to a limit".

The symbol :

$$\displaystyle\lim_{n\to\infty}x_n$$

is the traditional way to denote "the limit of the sequence" $(x_n)$ and, in spite of its "typographical complexity", it is only one symbol.

We can more simply omit the $n\to\infty$ part, which is redundant, being already "included" into the definition of sequence as an infinite list of real numbers, and write : $L=lim(x_n)$ in place of $x_n \to L$ and its definition will be :

$$L=lim(x_n) \ \ \iff \ \ \forall \epsilon > 0 \ \exists N \in \mathbb N \ \forall n \ge N \ \ |x_n - L| < \epsilon.$$

You can see the details in :

We have to note that the bi-conditional :

$$\displaystyle\lim_{n\to\infty}x_n=x\iff (n\to \infty \implies x_n \to x)$$

will not do as definition of limit.

If we consider a finite sequence : $(1,2,3,4,5,6,7,8,9,10)$, we have that $n\to \infty$ does not hold and thus the conditional : $(n\to \infty \implies x_n \to x)$ is satisfied for an $x$ whatever.

This means that we are licensed to call an $x$ whatever "the limit" of the above sequence.

• Thank you very much for the added portion. – user170039 Apr 9 '15 at 13:17

The symbolism "$x_{n} \to x$ as $n \to \infty$" reads "$x_{n}$ gets closer to $x$ as $n$ gets larger". But, to state precisely what we mean by saying that is an entirely different matter.

What is interesting here is that the sense of mathematical preciseness is NOT a pedantic one, so you would never see any mathematician insists on assigning to every symbol a "one-to-one" definition. A significant example is of course the sign $dx$ in an integral. Its meaning varies from context to context. So does the symbol $\infty$.

The essential point is that when one scribbles then he knows exactly what he is doing. If you take a look at some older texts, then you would find that some authors write $\lim_{n = \infty}$ freely. No harm was committed, heartedly.

A major purpose of a mathematical symbol may be mnemonic, suggestive, and convenient for a given goal. Note that in a poem, which is also a capsule of thoughts, the same word may have a different meaning at a different place. Mathematical symbols, to me, play the same role.

There seems no need to make symbols "logical", for insisting on that leads one to nowhere. For example, I do not see any virtual effect on differentiating "$= o(1)$" from "$\in o(1)$" for those who know exactly what they mean by writing the former...

• My friend also said a variant of the same feeling that there '...seems no need to make symbols "logical"' but when I told this to other friends, they told me that though they think that sometimes being emphatic about logic is redundant, they think that every mathematical symbol should have a definition in terms of existing mathematical symbols. For example, we need to define the symbol $\displaystyle\sum$ in terms of, say $+$ and $\cdot$ simply because it is not one of the primitive symbols of our number system. – user170039 Mar 29 '15 at 9:45
• I would say that concepts themselves are what really matter in mathematics. And, yes, mathematics defines new concepts in terms of older concepts! Symbols themselves are auxiliary. – Megadeth Mar 29 '15 at 9:48
• Whatever you want to write for the concept of convergence of a sequence, everybody eventually refers to the $\epsilon$-definition, rather than what you write. – Megadeth Mar 29 '15 at 9:51
• Actually my question was regarding the seemingly obvious paradox of the logical interpretation of $n\to\infty\implies x_n\to\infty$ and $x_n\to\infty$ which are not equivalent as @John said. But we assume that they are equivalent. – user170039 Mar 29 '15 at 9:54
• Please recall that $\infty$ is not a number if we are talking about the real numbers. We cannot operate with $\infty$! – Megadeth Mar 29 '15 at 9:57