[see also: decrease, diminish, reduce] The importance of these examples lay not only in lowering the dimension of known counterexamples, but also in...... [Note that the past tense of lie is lay, not “lied''.]
The lower limit is defined analogously: simply interchange sup and inf in (1).
The reader may wonder why we have apparently ignored the possibility of obtaining a better lower bound by considering......
a path obtained by going from A to B along the lower half of the circle
a lower semicontinuous function