Problem
If the sequence ${f_n}\subset C[a,b]$ is uniformly continuous, it is bounded.
If the sequence ${f_n}\subset C[a,b]$ is bounded, it is uniformly continuous.
Maybe someone can give me some hints for this? Thanks so much in advance!
2026-04-06 19:14:41.1775502881
Uniform continuity and bounded sequence
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1) Take a sequence of constant functions that diverge to infinity
2) Take a sequence $f_n (t) =[(b-t)(b-a)^{-1} ]^n $