Exercise: Show that if $f: [a,b] \to X$,a complete normed vector space, is Riemann integrable, then $f$ is bounded.
Definition of Riemann-integrable:
Let $f: [a,b] \to X$ be a map. F is Riemann-integrable if for every $\epsilon >0$ there exists a step function $u: [a,b]\to X$ and a step-function $v:[a,b]\to \mathbb{R}$ such that:
$(\forall x\in[a,b]: \| f(x)-u(x)\|\leq v(x))$ and $\mathbb{I}v<\epsilon$
I just don't see where to start with this one.
Am I supposed to (1) argue contra-positively or (2) show that if $\forall x\in[a,b]:\| f(x) – u(x)\|\leq v(x)$ and $v(x)$ can become arbitrarily small for $\forall x\in [a,b]$, then that must mean that $u(x)$ at some $x$ achieves the max value $M$, ($|f(x)| \leq M$) , that bounds $f(x)$?
Best Answer
If $f: [a,b] \to \mathbb{R}$ is Riemann Integrable then we have that $\inf_{P}U(f,P)=\sup_{P}L(f,P)$ where these are the lower and upper Darboux Sums, and P is a partition of $[a,b]$. For any particular partion of [a,b] we have that $U(f,P)=\sum\limits_{k=1}^{n}{(x_k-x_{k-1})M_k}$ where $M_k=\sup_{x_{k-1} \leq x \leq x_k}{f(x)}$. Suppose for sake of contradiction that f(x) is unbounded on [a,b] This implies that $\forall M > 0$ there exists $x \in [a,b]$ such that $f(x) > M/||P||$ where $||P||$ is the width of the partition.
Then consider for any partition P the Darboux sum will be bounded below by $||P||M/||P||=M \to \infty$. Hence, the Darboux sum will diverge and thus the function cannot be Riemann Integrable.