Why why why (tiny limsup proof)

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The discussion centers on a theorem regarding cluster points and the limits of sequences, specifically stating that if x is a cluster point of the sequence {x_n}, then lim inf(x_n) ≤ x ≤ lim sup(x_n). The author critiques the proof method presented in the book, suggesting a more straightforward approach by defining A as the set of all cluster points, leading to the conclusion that lim inf(x_n) = inf(A) ≤ x ≤ sup(A) = lim sup(x_n). This alternative proof is deemed more concise and direct.

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quasar987
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There's a "theorem" in my book that says if x is a cluster point of {x_n}, then lim inf(x_n)[itex]\leq[/itex]x[itex]\leq[/itex]lim sup(x_n).

The way the author proves it is a little bit extravagant. Why not just say "Let A be the set of all cluster points. Then, lim inf(x_n)=inf(A)[itex]\leq[/itex]x[itex]\leq[/itex]sup(A)=lim sup(x_n)."

?
 
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Don't they pretty much say the same thing?
 

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