[see also: decrease, diminish, reduce, cut down, limit]
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
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