- #1
jeckt
- 19
- 0
Homework Statement
Let [tex] \{ f_{n} \}_{n=1}^{\infty} \subset C[0,1] [/tex] be twice differentiable, and satisfying [tex] 0 = f_{n}(0) = f'_{n}(0) [/tex] and [tex] \| f''_{n}\|_{\infty } [/tex]. Prove that [tex] \{ f_{n} \}_{n=1}^{\infty} [/tex] has a convergent subsequence.
Homework Equations
So since [tex] C[0,1] [/tex] is a compact metric space. If [tex] \{ f_{n} \}_{n=1}^{\infty} \subset C[0,1] [/tex] is bounded and equicontinuous, then it has a convergent subsequence.
The Attempt at a Solution
I can show that it is bounded. The hint the lecturer gave us was to consider the remainder term of the taylor polynomial. That is what I used to show it is bounded. As for equicontinuous - I'm having a little difficult, I'm trying to prove it using the definition and the remainder term of the taylor polynomial.
Thanks for the help guys!