对于数列xn，其上极限为lim sup n→∞ xn 。
- For the series −xn, each term is the inverse of the xn counterpart.
- So, for any n, there is sup k≥n (−xk)=−inf k≥n xk (because negation changes the position of the maximum and minimum values, but keeps the difference between them constant).
- Similarly, the reversal of the upper and lower limits also holds.
It is evident from the diagram above that the negative value of the lower limit of the positive value function sequence xn is equal to the upper limit of -xn.
Therefore, the upper and lower limits of the sequence have the following properties:
In short, when considering the true bound of a function and its opposite, it can be found that their numerical values are equal in magnitude but opposite in sign, thus proving that the lower definite bound of xn is equal to the opposite of the upper definite bound of -f.
The following is Fatu's lemma:
Applying the above rules to the inverse figurine lemma:
In the proof, as long as fn is replaced with -fn on the basis of the fatu lemma, and the above properties of the upper and lower limits of the series are applied, the required conclusion can be obtained.