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Prove with an e – N argument that {fn(x) + gn(x)} converges uniformly to f (x) + g(x) on R.

Prove with an e – N argument that {fn(x) + gn(x)} converges uniformly to f (x) + g(x) on R.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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COURSE: Mathematical/Real Analysis (UC2)

TOPIC: Uniform Convergence 

Suppose that \(\{f_n(x)\}\) and \(\{g_n(x)\}\) are sequences of real-valued functions on \(\mathbb{R}\) such that \(\{f_n(x)\}\) converges uniformly to \(f(x)\) on \(\mathbb{R}\) and \(\{g_n(x)\}\) converges uniformly to \(g(x)\) on \(\mathbb{R}\). Prove with an \(\epsilon - N\) argument that \(\{f_n(x) + g_n(x)\}\) converges uniformly to \(f(x) + g(x)\) on \(\mathbb{R}\).
Transcribed Image Text:Suppose that \(\{f_n(x)\}\) and \(\{g_n(x)\}\) are sequences of real-valued functions on \(\mathbb{R}\) such that \(\{f_n(x)\}\) converges uniformly to \(f(x)\) on \(\mathbb{R}\) and \(\{g_n(x)\}\) converges uniformly to \(g(x)\) on \(\mathbb{R}\). Prove with an \(\epsilon - N\) argument that \(\{f_n(x) + g_n(x)\}\) converges uniformly to \(f(x) + g(x)\) on \(\mathbb{R}\).
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