I need to prove the following: Let $f(x)$ be a continuous function in $[a,b]$.
Prove that every partition $T:a=x_0<x_1<...<x_n=b$ has a Riemann sum: $\sigma=\sum_{i=1}^{n}f(\alpha_i)\Delta x_i$ (when $\alpha_i\in(x_{i-1},x_i)$), and $\sigma = \int_{a}^{b}f(x)dx$.
I started proving with induction: When $n=1$, you can simply use the Mean Value Theorem, and you get that $\int_{a}^{b}f(x)dx=f(c)(b-a)$ for a $c\in(a,b)$, and you can take $c$ into the Riemann sum.
However, I can't find a way to prove that if it is true for $n$, it's true for $n+1$ as well. Help please!
Thanks.
There is no induction needed; Using what we proved for $n=1$, we know that for every interval $[x_{i-1}, x_i]$, $\int_{x_{i-1}}^{x_i}f(x)dx=f(c_i)(x_{i-1}-x_i)$ for a $c_i\in(x_{i-1},x_i)$. When taking all $c_i, i=1,..,n$ into the Riemann sum you get:
$\sum_{i=1}^{n}f(c_i)(x_i-x_{i-1}) = \sum_{i=1}^{n}\int_{x_{i-1}}^{x_i}f(x)dx=\int_{a}^{b}f(x)dx$
Thank you @Dirk for explaining this to me!