= Supremum can fail to be attained at an oscillatory endpoint
The <extreme value theorem> requires continuity on the entire closed interval. A bounded function can have values tending to an endpoint envelope without ever reaching that envelope, while its defined endpoint value is smaller. To prove nonattainment, establish a strict pointwise upper bound and construct a sequence approaching that bound. An oscillatory factor can supply the approaching sequence by selecting phases where it equals $1$.
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