Suppose $\displaystyle f $ is a real valued function whose domain contains the interval $\displaystyle [a,b] $. Assume that $\displaystyle f $ is unbounded on $\displaystyle [a,b] $.

(a) Prove that there exists a sequence $\displaystyle (y_n) $ in $\displaystyle [a,b] $ that converges to $\displaystyle y \in [a,b] $ such that for every $\displaystyle n \in \mathbb{N} $, $\displaystyle |f(y_n)| > n $.

Isn't this just the definition of an unbounded sequence? Do we have to actually construct one? What would be the easiest way to do this?

(b) Using this unbounded sequence how would we show that the set of Riemann sums of $\displaystyle f $ corresponding ot $\displaystyle P $ is an unbounded set of real numbers?