I am following "Concrete Mathematics"'s treatment of Harmonic Numbers (pg. 262 onwards). They show that $\lim_{n\to\infty}\left(H_{n}-\ln n\right)=1-\frac{1}{2}\left(\zeta\left(2\right)-1\right)-\frac{1}{3}\left(\zeta\left(3\right)-1\right)-\dots$ using some beautiful arguments, and then claim the right hand side converges (and is the famous Euler's constant). However, I don't understand how they can claim it converges without further justification, which I can't find.
Indeed, they show just before that $\ln n