Are the limit points of a sequence and of a set defined differently? Are limit points of a sequence and an interval differently defined? I ask because I've been given the following definition of a limit point:


  A limit point is one such that every of its neighborhoods contains infinitely many points of the set other than itself.


If that is so then this criteria applies only when we talk about an interval and it fails to give limit points of any sequence - for instance $(-1)^n$ has limit points at $1$ and $-1$, but these are not limit points according to the above definition.

What's the discrepancy between the limit points of a sequence and the definition given above?
It's true that, taken naively, this definition does conflict with that of the limit points of a sequence. The problem is that one is counting the values $1$ and $-1$ only once, whereas they occur infinitely many times in the sequence. That is to say that we really should distinguish between the set $\{1,-1\}$ (which has no limit points) and the sequence $(-1)^n$ (which has two limit points).

A more analogous statement for sequences would be:


  A limit point is a point such that the sequence has infinitely many terms in any neighborhood of it.


Which just basically repairs the statement for sequences by changing "distinct terms" to just "terms" - so that factors which are repeated infinitely often get included as limit points. So, all we need to do is be a bit careful about how we could when we have a set vs. how we count when we have a sequence.
A limit point of a sequence $(a_n)_{n\to \infty}$ is defined as the point the sequence itself gets close. An interesting example of this is the sequence $(1)_{n\to \infty}$ approaches $1$. 

While the limit point of a set is seen as a point in which every neighborhood of that point (the limit-point) contains points other than itself. More interesting the set $\{1\}$ has no limit point and the set, the set $\{1+\frac{1}{n}: n\in \mathbb{N}\}$ has $1$ as it's only limit point, and the set $\mathbb{R}$ has itself has it's set of limit points.

Conceptually they hold the same idea of closeness to a point. But mechnically they work a bit different. Sequences can only have one limit point (if we are talking about metric spaces), while Sets can have an infinite set limit points.