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Let {f} be a sequence of continuous functions which converges uniformly to a function f on a set SC R. Let {xn} ≤ S be a sequence converging to x € S. Prove that lim fn(x) = f(x). n→∞ Is this statement true if we assume fn → f pointwise? Why?

Let {f} be a sequence of continuous functions which converges uniformly to a function f on a set SC R. Let {xn} ≤ S be a sequence converging to x € S. Prove that lim fn(x) = f(x). n→∞ Is this statement true if we assume fn → f pointwise? Why?

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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Let {f} be a sequence of continuous functions which converges uniformly to a function f on a set SCR. Let {n} S be a sequence converging to x E S. Prove that lim fn(x) = f(x). n- Is this statement true if we assume fn → f pointwise? Why?
Transcribed Image Text:Let {f} be a sequence of continuous functions which converges uniformly to a function f on a set SCR. Let {n} S be a sequence converging to x E S. Prove that lim fn(x) = f(x). n- Is this statement true if we assume fn → f pointwise? Why?
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