Problem 3

Let

1=a_0\le a_1\le a_2\le\cdots

be a sequence of real numbers. Define

b_n=\sum_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac1{\sqrt{a_k}}

. (a) Prove that

0\le b_n<2

for every

n

. (b) Given any

b

with

0\le b<2

, prove that there is such a sequence for which

b_n>b

for infinitely many

n

Step 5 of 5: Reach every target below

2

Previous step Next step

In plain words

A longer interval approaches area $2$ , and a finer mesh makes the sum follow it.

\\lim_{n\\to\\infty}\\left(2-\\frac2{\\sqrt{1+n\\Delta x}}\\right)=2

Detailed analysis

Choose a small $\Delta x>0$ . As $n\to\infty$ , the integral tends to $2$ . For $b<2$ , first take $n$ large enough that the integral exceeds $b$ , then make the mesh fine enough that the Riemann sum still satisfies $b_n>b$ ; this gives infinitely many $n$ .