Editing Constructive analysis (section)

===The least-upper-bound principle and compact sets===
Another difference between classical and constructive analysis is that constructive analysis does not prove the [[least-upper-bound principle]], i.e. that any [[subset]] of the real line '''R''' would have a [[least upper bound]] (or supremum), possibly infinite.
However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any ''located'' subset of the real line has a supremum.
(Here a subset ''S'' of '''R''' is ''located'' if, whenever ''x'' < ''y'' are real numbers, either there exists an element ''s'' of ''S'' such that ''x'' < ''s'', [[logical disjunction|or]] ''y'' is an [[upper bound]] of ''S''.)
Again, this is classically equivalent to the full least upper bound principle, since every set is located in classical mathematics.
And again, while the definition of located set is complicated, nevertheless it is satisfied by many commonly studied sets, including all [[Interval (mathematics)|intervals]] and all [[compact set]]s.

Closely related to this, in constructive mathematics, fewer characterisations of [[compact space]]s are constructively valid—or from another point of view, there are several different concepts that are classically equivalent but not constructively equivalent.
Indeed, if the interval [''a'',''b''] were [[sequentially compact]] in constructive analysis, then the classical IVT would follow from the first constructive version in the example; one could find ''c'' as a [[cluster point]] of the [[infinite sequence]] (''c''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub>.