Difference between limit point and limit

Question

What is the difference between limit point of a sequence and limit of a sequence. Can it be unique?

Answer 1

Just one more point,

Limit Point is defined for set.

Limit is used in context of sequence.

Answer 2

Gaurav Please look at the definitions and tell what trouble you are facing in understanding the definitions.

• http://en.wikipedia.org/wiki/Limit_point

• http://en.wikipedia.org/wiki/Limit_of_a_sequence

Answer 3

I believe he may be referring to Kreyszig Functional Analysis, in which the definition of limit point does not match that in wikipedia, and looks very close (identical?) to what he states as the definition of limit.

Answer 4

Depend of sequence. For example, if you have a sequence of numbers, for example if it is a sequence of real numbers only make sense one limit type, in this case "limit point" is equal to the "limit". On the other hand if you have a sequence of functions (for example) $(f_n)_{n \in \mathbb{N}}$, $f_n:X \to \mathbb{R}$, $X$ Banach space (for example), you have two limit's type.

Limit of the sequence: $$\lim_{n\to \infty}f_n=g$$ if for each $\varepsilon>0$, exist $n_0 \in \mathbb{N}$ such that $$n>n_0 \Rightarrow \|f_n-g\|=\sup_{x\in B(0,1)}|f_n(x)-g(x)|<\varepsilon$$

Limit point: $$\lim_{n\to \infty}f_n(x_0)=g(x_0)$$ if for each $\varepsilon>0$, exist $n_0 \in \mathbb{N}$ such that $$n>n_0 \Rightarrow |f_n(x_0)-g(x_0)|<\varepsilon.$$