Supremum and infimum

• How to Prove That √2 Is the Supremum of {x ∈ ℝ : x² < 2}
  Prove that √2 is the least upper bound of all real x satisfying x² < 2, and see why it is a supremum but not a maximum.
• How to Prove Inequalities for Infima and Suprema
  The title does not specify an inequality, so this article proves a standard one: the difference of two infima is bounded by the supremum of the pointwise difference.
• How to Prove That a Monotone Bounded Sequence Converges
  A real monotone sequence converges if it is bounded in the direction it moves. Here is the supremum proof, an example, and common pitfalls.
• How to Prove That sup(A ∪ B) = max{sup A, sup B}
  If A and B are nonempty and bounded above, the supremum of their union is the larger of their individual suprema. Here is the least-upper-bound proof.
• What Are the Minimum, Maximum, Infimum, and Supremum?
  Learn how minimum, maximum, infimum, and supremum differ, with definitions, examples, and a test for whether a bound is attained.
• How to Find the Infimum and Supremum of a Riemann Sum
  For a fixed partition, the infimum and supremum of tagged Riemann sums are the lower and upper Darboux sums; varying partitions changes the answer.
• Find the Infimum and Supremum of the Set
  Separate the even and odd cases of n to find the exact bounds: the set has minimum −1 and maximum 2, so its infimum and supremum are attained.
• Why Does a Nonempty Bounded Set of Reals Have a Supremum?
  A nonempty set of real numbers that is bounded above has a supremum because the real numbers satisfy the least-upper-bound property.
• What Is the Difference Between Supremum and Maximum?
  A supremum is the least upper bound and need not belong to a set; a maximum is the greatest element, so every maximum is a supremum but not vice versa.
• Find the Supremum, Infimum, lim sup, and lim inf of 2 − 1/n
  For n ≥ 1, the sequence 2 − 1/n has infimum and minimum 1, supremum 2, and both lim sup and lim inf equal to 2.
• Does Every Nonempty Bounded-Above Set of Real Numbers Have a Supremum?
  Every nonempty subset of the real numbers that is bounded above has a supremum, by the completeness property of the real numbers.