- #1

Euklidian-Space

- 38

- 0

## Homework Statement

Suppose that ##f_{n} \rightarrow f## uniformly on [a,b] and that each ##f_{n}## is integrable on [a,b]. Show that given ##\epsilon > 0##, there exists a partition ##P## and a natural number ##N## such that ##\left|L(f_{n}, P) - L(f,P)\right| < \epsilon##.

## Homework Equations

## The Attempt at a Solution

I let P be a partition. And

$$m_{k} = inf\{f(x) : x \in [x_{k-1},x_{k}\}$$

$$m_{k}^{'} = inf\{f_{n}(x): x \in [x_{k-1}, x_{k}]\}$$

I am thinking maybe I can say ##|m_{k} - m_{k}^{'}| \leq \frac{\epsilon}{b - a}##, but i do not really know how to justify it. formally anyway