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 $n$

This answer may be off-topic; so, please, forgive me if this is the case.
$$\sum_{i=1}^n\frac 1i=H_n$$ the rhs being the harmonic number. For large values of $n$, the asymptotic expansion is $$H_n=\gamma +\log (n)+\frac{1}{2 n}-\frac{1}{12
   n^2}+O\left(\frac{1}{n^3}\right)$$ This makes $$a_n=\sum_{i=1}^n\frac 1i-\log(n)=\gamma +\frac{1}{2 n}-\frac{1}{12
   n^2}+O\left(\frac{1}{n^3}\right)$$
