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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_05” 2011/5/22 22:55 page 36 #36
5-36 Real Analysis
Thus we have obtained the required inequality for θ ∈[0, π]. Now let
θ ∈[π,2π]. Put φ = θ π so that 0 φ π and
|sin θ|=|sin φ|≤|φ|=φ = θ π θ =|θ |.
If θ ∈[0, ∞]\[0, 2π], we write φ = θ 2nπ for a suitable positive
integer n 1 so that φ ∈[0, 2π ]. Again we have
|sin θ|=|sin φ|≤|φ|=φ = θ 2nπ θ =|θ |.
Hence the inequality is valid for θ ∈[0, ). Again changing θ to θ
we
find that the above inequality is valid for θ (−∞,0] and hence
for all real θ.
5.8 GENERALIZATIONS
Most of the definitions and results that we have obtained in this chapter
for real-valued functions of a real variable can be generalized to the
context of a metric space with slight modifications wherever ...
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Publisher Resources

ISBN: 9781299447561Publisher Website