# Proving the inequality

relinquished™
I encountered this problem in one of my math lecture notebooks; Our professor didnt show how it was done, so that leaves me clueless. The problem was to show that the sequence {a_n} defined by
$$a_1 = 1, a_2 = \int^2_1 \frac{dx}{x}, a_3 = \frac{1}{2}, a_4 = \int^3_2 \frac{dx}{x} , ...$$

When generalized gives For any natural number n,

$$a_{2n-1} = \frac{1}{n}$$
$$a_{2n} = \int^{n+1}_n \frac{dx}{x} = \ln x |^{n+1}_{n} = \ln \frac{n+1}{n}$$

is decreasing, that is,
$$\frac{1}{n} > \ln \frac{n+1}{n} > \frac{1}{n+1}$$

I've tried math induction but I'm stuck at the (ii) part of math induction, and i tried comparing their derivatives, but I can't conclude anything from doing so. I've tried to compute for their areas, but that got me nowhere. I've graphed their functions using a graphing program, and I saw that it is true, but I would like know how i can prove this without graphing...

Homework Helper
when a<b
$$(b-a)\min(f)\leq\int_a^b f dx\leq(b-a)\max(f)$$