supN

SupN, written as sup_N a_n or sup_{n∈N} a_n, denotes the supremum (least upper bound) of a set of real numbers indexed by the natural numbers. Given a sequence (a_n), one often considers the set S = {a_n : n ∈ N}. The supremum is the smallest real number that is greater than or equal to every term of the sequence. If no finite upper bound exists, the supremum is +∞ in the extended real numbers. If there exists an index n0 with a_{n0} = sup_N a_n and a_n ≤ a_{n0} for all n, then the supremum is attained and equals the maximum of the sequence.

Key properties include that sup_N a_n is an upper bound for all terms and is the least such bound. For any ε > 0, there exists n with a_n > sup_N a_n − ε, illustrating that the supremum is the best possible bound from above, though not every sequence attains it. If the sequence has a maximum (i.e., some a_{n0} is at least as large as every other term), then sup_N a_n equals that maximum.

Examples: For a_n = (-1)^n, the set of values is {−1, 1}, so sup_N a_n = 1. For a_n = 1/n, the set is {1, 1/2, 1/3, ...}, so sup_N a_n = 1. For a_n = −n, the supremum is −1, attained at n = 1.

Sup_N is a fundamental concept in real analysis and is related to, but distinct from, the maximum and from related ideas such as the limit superior (limsup).