Preliminaries
Let $(x_n)$ be a bounded real number sequence and $ (x_n )_{n≥k} $be a subsequence of $x_n$ which only takes the values of the sequence starting from the k−th term.
Let {$x_n $} and {$x_n$ }$_{n≥k}$ denote a subset of R that contains all values of the corresponding sequences $(x_n )$ and $(x_n )_{n≥k}.$
Initial Considerations
Define two sequences ($\alpha_k$) and ($\beta_k$) out of the original ($x_n$) sequence :
($α_k$)= sup{$x_n$}$_{n≥k}$
($\beta_k$)= inf{$x_n$}$_{n≥k}$
Call a number $c \in R$ a subsequential limit of the real number sequence ($x_n$) iff there exists a subsequence of ($x_n$) that converges to c.
Define C $\in R$ as the set of all subsequential limits of ($x_n$).
Definition of Lim sup and Lim inf
1 - Define lim sup $x_n$ as the supremum of C and define lim inf $x_n$ as the infimum of C.
2 - Define lim sup $x_n$ as the infimum of {$\alpha_k$} and lim inf $x_n$ as the supremum of {$\beta_k$}.
My doubt
Why the supremum of C is the same as the infimum of {$\alpha_k$} and the infimum of C is the same as the supremum of {$\beta_k$} ?
In another words, what's special about the construction of the ($\alpha_k$) and ($\beta_k$) sequences that make their limits ( their infimums/supremums because they are bounded monotone decreasing/increasing, respectively ) to be largest/smallest possible subsequential limits of the original sequence sequence ($x_n$) ?
Can i understand the equivalence between these two definitions ( preferably both intuitively and rigorously ) without touching on anything related to topology ?
The reason is that my real analysis teacher ( who has not taught us anything about topology yet ) is teaching us sequences using the second definition of lim sup and lim inf and many of the theorems we are going through are better understood ( in my view ) if one understands both definitions of lim sup and lim inf ( and obviously, their equivalence ).
Thanks a lot in advance.