Supremum and infimum
The supremum and infimum of sequences and of sets given by inequalities — and how the least upper bound differs from the largest element.
How to solve it
The least upper bound ($\sup$) is the smallest number not less than every element of a set. It may belong to the set (then it is the largest element) or not: the interval $(0,\ 1)$ has supremum $1$ but no largest element. Every non-empty set of reals bounded above has a supremum — that is the completeness axiom.
Step by step
- A sequence: check whether it is monotonic. For a decreasing one the supremum is the first term and the infimum the limit (if it is not attained).
- A set from an inequality: solve it and look at the ends of the interval. A strict inequality leaves the endpoint out of the set, but it is still the bound.
- Split a complicated sequence into subsequences (say even and odd indices) and find the bounds of each.
- To check that $M$ is the supremum: all elements are $\le M$, and for every $\varepsilon > 0$ some element exceeds $M - \varepsilon$.
Common mistakes
- Thinking the supremum must belong to the set. $\{x : x^2 < 8\}$ has supremum $\sqrt{8}$, though $\sqrt{8}$ itself is not in it.
- Confusing “the largest element” with “the supremum”: the first may not exist, the second always does for a bounded set.
- Taking the limit as a bound without checking monotonicity: an oscillating sequence may have other bounds.
- Looking for the bound of a set of rationals among the rationals — it can be irrational.