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 3 of 5: Bound by the integral

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In plain words

The total area can approach the limiting value $2$ but does not reach it at a finite endpoint.

0\\le b_n<\\int_1^{a_n}x^{-3/2}dx=2-\\frac2{\\sqrt{a_n}}<2

Detailed analysis

Every summand is nonnegative. The rectangles lie below $f$ on the interval from $1$ to $a_n$ , so their total is less than the integral, giving the stated bound.