Subsequences Converging to Limsup and Liminf of a Sequence

We construct subsequences of a sequence which converge to the limit superior and limit inferior respectively. The supremum approximation property allows us to find a terms in the original sequence which are as close as we wish to the supremum or infimum past a certain point in the sequence; this is the essential part of the proof. #mikethemathematician, #mikedabkowski, #profdabkowski