- #1
jdz86
- 21
- 0
Homework Statement
Suppose f:[a,b] [tex]\rightarrow[/tex] [tex]\Re[/tex] is bounded and that the sequences {[tex]U_{P_{n}}[/tex](f)}, {[tex]L_{P_{n}}[/tex](f)} are covergent and have the same limit L. Prove that f is integrable on [a,b].
Homework Equations
[tex]U_{P_{n}}[/tex](f) is the upper sum of f relative to P, and [tex]L_{P_{n}}[/tex](f) is the lower sum of f relative to P, where in both cases P is the partition of [a,b].
Also a hint was to examine the sequence {[tex]a_{n}[/tex]} where [tex]a_{n}[/tex] = [tex]U_{P_{n}}[/tex](f) - [tex]L_{P_{n}}[/tex](f)
The Attempt at a Solution
Didn't figure out the full proof, only parts:
I started with having P = {[tex]x_{0}[/tex],..., [tex]x_{n}[/tex]}, where [tex]x_{0}[/tex] = a, and [tex]x_{n}[/tex] = b. And given that they are both convergent and have the same limit, using the hint I can show that they both converge to [tex]a_{n}[/tex], but I got stuck after that.