Proving monotonicity and convergence of a sequence $a_n=(1+{1 \over 2}+{1 \over 3}+....+{1 \over n})-\ln(n)$
Show that $a_n$ is bounded and monotone and hence convergent. 
I know that the $-\ln(n)$ portion will be monotonically decreasing. I think I need to somehow show that $1+{1 \over 2}+{1 \over 3}+....+{1 \over n}>\ln(n)$ so that the whole sequence will be monotonically increasing ie. $a_n<a_{n+1}$ and also need to show that for all $n$ there is a $C$ such that $a_n<C$ for all $n$

There is a nice "graphical proof" of the convergence of the (increasing) parent sequence
$$1-a_n=\ln(n)-\left({1 \over 2}+{1 \over 3}+....+{1 \over n}\right)$$
towards a limit: move the different involved surfaces into the unit square. As they do not overlap, the blue sawtooth region converges to a limit region whose area, slightly smaller than $1/2$, is $1-\gamma \approx 0.423...$, where $\gamma$ is Euler's constant.
 
