As we all know, the sequence $1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n$ is decreasing and bounded. **Its limit is called Euler constant denoted by $\gamma$**.
I can get $$0<1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n<1.$$
And if we use the limit $\gamma=0.577~215\cdots$,
we know 
$$1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n>\frac{1}{2}.$$
Now my question is how to prove 
$$1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n>\frac{1}{2}$$
without the properties of limit and $\gamma=0.577~215\cdots$.