Show that limsupn →∞(-an)=-lim infn →∞ an.

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Key Concepts

Negation and Order Reversal in Suprema and Infima

When a set of real numbers is negated, the supremum and infimum of the set are interchanged with a sign change. Specifically, the negation of the supremum of a set equals the infimum of the negated set, and vice versa. This property is critical in the given problem as it links the limit superior of the negated sequence to the negative of the limit inferior of the original sequence.

Limit Inferior

The limit inferior of a sequence, denoted as lim inf, is the smallest accumulation point of the sequence or the limit of the infimum of the tails of the sequence. It serves as the eventual lower bound of the sequence, capturing the minimum values that the sequence approaches repeatedly. The concept is central in real analysis for assessing the long-term minimal tendencies of sequences that may not converge to a single value.

Limit Superior

The limit superior of a sequence, often denoted as lim sup, is the largest accumulation point of the sequence or the limit of the supremum of the tails of the sequence. It reflects the eventual upper bound of the sequence, even if the sequence does not converge. In our context, it helps quantify the long-term maximal behaviour of the sequence, which is crucial for analyzing sequences that oscillate or do not possess a regular limit.

Show that $\lim \sup _{n \rightarrow \infty}\left(-a_n\right)=-\liminf _{n \rightarrow \infty} a_n$.

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