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

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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.
 
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ZaidAlyafey said:
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.
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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