Show that the constant function is integrable and find its value of integration.
Suppose such that where is any constant. Let be any partition on , ie then Upper Darboux sum and Lower Darboux sum we evaluate by Now what about the supremum and infimum of ? If but then is constant so infimum of is also which immediately follows that Further shows that is integrable and its values is
Now how to show that the supremum and infimum of the constant function is constant itself without using completeness property?
Any sorts of help will be appreciated.