Are sequences and sets the same thing?

[Damidami]

I think I'm not understanding something here:
A point $L \in \mathbb{R}$ is a limit point of a sequence $a_n$ if exists a subsequence $b_n$ such that $\lim b_n = L$

So for example the constant sequence $a_n = 1$ so that $a = 1, 1, 1, 1, 1, 1, \ldots$ has a unique limit point $L=1$

But a limit point (or acumulation point) is one that can be approached by nearby point in the set. (For example in the open interval $(0,2)$ we have that 2 is al limit point, but in the set $S=\{ 1 \}$ we have no limit point (1 is an isolated point in $\mathbb{R}$)

Aren't both definitions of limit point contradictory? What am I doing wrong?

Thanks

 

[alexfloo]

First of all, they are not definitions of the same thing. The limit point of a sequence (also called a subsequential limit) is different from a limit point of a set.

However, they are closely related. Your confusion stems from the fact that 1 is in fact a limit point of the set ${1}$. The constant sequence a_n=1 is a sequence in the set which approaches 1. (Yes, we actually do say that a constant sequence "approaches a limit" even though it's already there.)

In general, every point in a set is a limit point of that set. The limit points of the sequence $a_n$ are a subset of the limit points of the set (also called the closure of the set) ${a_n}$.

 

[Damidami]

Thank you alexfloo, but something still doen't make sense to me.

Let's consider the set $S = \{1\}$ as a subset of $\mathbb{R}$.
Then the element $1 \in S$ is not a limit point, but instead is an isolated point (there are not other points of the set $S$ "near" the element $1$).

On the other hand, the limit point of a sequence need not be a point of the sequence (considered as a set), for example $a_n = 1/n$ has $0$ as a limit point, but there does not exist $n_0 \in \mathbb{N}$ such that $a_{n_0} = 0$

I am wrong on something?

That said, I think you are true that there are two "non-compatible" definitions of limit points: one for sequences, and other for sets: A limit point of a sequence is not necesarily a limit point of the same sequence considered as a set.

Please correct me if I'm wrong.

 

[alexfloo]

You are absolutely correct, the definition I game of the limit point of a set was wrong.

The limit points of a sequence need not be a limit point of the set of elements of that sequence. However, the subsequential limits of a sequence must be either a limit point or an isolated point.

The definitions are slightly different, but that's okay: they're *definitely* not contradictory, since they define different things.

 

[Damidami]

What confuses me is that sometimes sequences are thinked as sets, for example given
$c_n = 1,1,1,1,1,1,1,\ldots \ldots$
$d_n = 1,2,1,2,1,2,1,2,\ldots$
(repeating the pattern)

Then I have seen written things like
$\{c_n\} \subset \{d_n\}$

But when we talk about limit points of a sequence, it is usually think as a function and not as a set, that is
$c : \mathbb{N} \to \mathbb{R}$
represents a function. (And the related set I think is the image of that function)

 

[alexfloo]

That notation can be confusing, but it's important to recognize that those sequences are not sets.

${c_n}$ is actually shorthand for ${c_n:n\in\mathbb{N}}$, or "the set of all terms of the sequence," not just "the sequence.

The notation: $\{c_n\}\subset\{d_n\}$ doesn't mean "one sequence is a subset of another," it means "the set of terms of one sequence is a subset of the set of terms of the other."

We can always take a sequence and turn it into a set, (and it's often really helpful to do that) but we can't get the sequence back, and so they're not the same thing.

As for your last point, that's exactly correct. Sequences are functions, and the set $\{c_n\}$ is the image of that function.