“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 ...