Question

Task 16. For a continuous function f, let lfll axeo.f(x) Prove that continuous functions fn uniformly converge to a continuous function f if and only if limn + In f-
PLEASE use the THEORY below to give PROOF STEP BY STEP. This is an analysis class. Thank you.

• application of power series\Weierstrass M-test\term by term differentiability of power series
• sequence and series of function: pointwise and the theorem of uniform convergence
• which function is integrable: continuous and monotone
• Fri 19 Apr: The Fundamental Theorem of Calculus. (§7.5.)
• Wed 17 Apr: Example (∫10x2dx=1/3∫01x2dx=1/3). Basic properties of the integral. (mostly Theorem 7.4.2.)
• Fri 12 Apr: More on integrability, basic properties of the integral. (§§7.3–7.4.)
• Wed 10 Apr: Integrability. Integrability criterion. Continuous functions are integrable. (§7.2)
• Mon 8 Apr: Introduction to integration. Partitions. (§§7.1–7.2)

Answer 1

Fimetiony and eonvenges um