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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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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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