of the last inequality is driven to equal the right side in the limit and is sandwiched in between. This can only mean that In the context of this proof this means that has the three specified properties belonging to . Also, the proof provides a specific expression for . And the final critical part of the proof is to remember that the limit of a sequence is unique. This means that for any choice of only one possible number can exist. Therefore, there is no other function with all the properties assigned to . The remaining loose end is the question of proving that makes sense for all where exists. The problem is that our first double inequality was constructed with the constraint . If, say, then the fact that is monotonically increasing would make , contradicting the inequality upon which the entire proof is constructed. However, which demonstrates how to bootstrap to all values of where the limit is defined. See also Wielandt theorem References Gamma and related functions Theorems in complex analysis