For each positive integer n, let
{y_n}= 1 + 1/2 +...+1/n- the integral of (1/x) from 1 to n.
Prove that the sequence {y_n} converges.
Proof of monotone decreasing:
To prove that is bounded below by , if is the partition , and , the upper sum of the integral of is . Since , it follows that for all .
Since is a monotonically decreasing sequence bounded below, it converges.
This is actually a famous limit and converges to the Euler-Mascheroni Constant, .
You're welcome.