Basic Real Analysis by Houshang H. Sohrab

Author:Houshang H. Sohrab
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Proof.

Let be given. Since ϕ is continuous, it is bounded and uniformly continuous on [m, M]. Therefore we have | ϕ(s) | ≤ K for some K > 0 and all s ∈ [m, M], and we can find δ > 0 such that for all s,  t ∈ [m, M] satisfying | s − t |  < δ. Also, since we can pick a partition of [a, b] such that

(*)

Let m j (resp., M j ) be the infimum (resp., supremum) of f on [x j−1, x j ] and let m j ′ and M j ′ be the corresponding numbers for g. Divide the set {1,  2, …,  n} into two subsets:

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.