Proof of Existence: IVP w/ Continuous I & b in I

[onie mti]

i was given that f is a real alued function defined on an open interval I with IVP
x'(t) = f(x(t)) where x(s) = b

how would I go to prove that if I is continuous on I and b is in I then there exists a positive number say k and a solution x for the initial value problem defined on (s-k,s+k)

 

[alexmahone]

This is the Cauchy-Peano Existence Theorem.

The statement and proof are on http://www.math.unl.edu/~s-bbockel1/933-notes/node1.html.