proof that a function is integrable on a interval $[a,b]$ a) Divide a interval $[a,b]$ into $n$ equal subintervals. 


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*here I'm thinking $P_{n} =(x_0,x_1,x_2,x_3,x_{n-1}, x_n)$ where $a = x_0 < x_1 < x_2 < x_3 <\dots< x_{n-1} < x_n = b$ 


b) make an expression for the lower Riemann sum $L(f,P_{n})$ and the upper $U(f,P_{n})$. 


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*here I also got an idea, what to do. The hardest part is the last 4 assignments:


c) show that $U(f,P_n) - L(f,P_n) = \frac{(b-a)(f(b)-f(a))}{n}$
d) show that for a given $\epsilon>0$ calculate a $n'$ so that $U(f,P_{n'}) - L(f,P_{n'}) < \epsilon$
e) use the result from d) to show that there is only one number $I$ which obeys $U(f,P_n) ≥ I ≥ L(f,P_n) \quad\forall n \in \mathbb{N}$
f) Show that $f(x)$ is integrable on $[a,b]$
Where do you use that $f$ is continuous (if you demand this at all)?
I don't know what to do what the last 4 assigments. 
Kind regards Jones 
 A: As you are asked to specifically use equal subintervals, you can write explicitly that $x_i=a+i\frac{b-a}n$, $0\le i\le n$.
By definition, 
$$L(f,P_n)=\sum_{i=1}^n(x_i-x_{i-1})\inf\{f(x)\mid x_{i-1}\le x\le x_i\}$$
and
$$U(f,P_n)=\sum_{i=1}^n(x_i-x_{i-1})\sup\{f(x)\mid x_{i-1}\le x\le x_i\}.$$
Assuming that the function $f$ is non-decreasing (a property of $f$ that is not mentioned in your problem statement, but matches well the expected results), we know that $\inf\{f(x)\mid x_{i-1}\le x\le x_i\}=f(x_{i-1})$ and $\sup\{f(x)\mid x_{i-1}\le x\le x_i\}=f(x_i)$. Additionally using $x_i-x_{i-1}=\frac{b-a}n$, we obtain
$$\begin{align}U(f,P_n)-L(f,P_n)&=\frac{b-a}n\sum_{i=1}^nf(x_i)-\frac{b-a}n\sum_{i=1}^nf(x_{i-1})\\&=\frac{b-a}n\sum_{i=1}^nf(x_i)-\frac{b-a}n\sum_{i=0}^{n-1}f(x_{i})\\&=\frac{b-a}n(f(x_n)-f(x_0))=\frac{(b-a)(f(b)-f(a))}{n}\end{align}.$$
In the light of c it suffices to choose $n'>\frac{(b-a)(f(b)-f(a))}{\epsilon}$.
As the existence of at least one such $I$ is not asked for, we need only assume that there exist two such numbers $I_1< I_2$ and find a contradiction by letting $\epsilon=I_2-I_1$ in d.
Your definitin of integrability should be something like a restatement of e (however, does it really make use only of equidistant partitions?).
We did not make use of continuity, but only that $f$ is non-decreasing. (To repeat: I assume that the fullproblem statement includes this property of $f$)
