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Living Laser

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I think what was particularly confusing was a definition which defined lim sup as (sorry, new to the forum and still working out how to do subscript properly) inf(sub n>1) sup { [tex]A_{}n[/tex], [tex]A_{}n+1[/tex], [tex]A_{}n+2[/tex], ....}

The uniqueness of sup makes this definition seem meaningless to me. Obviously misinterpreting something dramatically here but hopefully someone can set me straight! Thanks.