What is the definition of lim sup and how is it related to subsequential limits?

[alyafey22]

Let $s_n$ be a sequence of real numbers and define $E$ to be the set of all subsequential limits of $s_n$ in the real extended line. Then we define the following

$$\lim \text{sup } s_n = \text{sup } E$$​

For some reason I don't quite understand the above formula , do we need to prove it ? It would be nice if you give some examples.

 

[Plato2]

Let $s_n$ be a sequence of real numbers and define $E$ to be the set of all subsequential limits of $s_n$ in the real extended line. Then we define the following

$$\lim \text{sup } s_n = \text{sup } E$$​

For some reason I don't quite understand the above formula , do we need to prove it ? It would be nice if you give some examples.

We really don't prove a definition. There are many equivalent ways to define lim sup.
But there is no point in reinventing the wheel. Have a look at this page.