MathDB

Let

f: [0,1]\rightarrow(0,+\infty)

be a continuous function. a) Show that for any integer

n\geq 1

, there is a unique division

0=a_{0}<a_{1}<\ldots<a_{n}=1

such that

\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx

holds for all

k=0,1,\ldots,n-1

. b) For each

n

, consider the

a_{i}

above (that depend on

n

) and define

b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}

. Show that the sequence

(b_{n})

is convergent and compute it's limit.

Problem Statement