Lebl-Basic-Analysis-I · Problem 7.6.7

Metric Spaces

Mathematics · picards-theorem

Problem

Consider

y' = y^2

y(0) = 1

. Use the iteration scheme from the proof of the contraction mapping principle. Start with

f_0(x) = 1

. Find a few iterates (at least up to

f_2

). Prove that the pointwise limit of

f_n

\dfrac{1}{1-x}

, that is, for every

x

with

|x| < h

for some

h > 0

, prove that

\displaystyle\lim_{n\to\infty} f_n(x) = \dfrac{1}{1-x}