TITLE: Showing inequality for harmonic series.
QUESTION [13 upvotes]: I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.

REPLY [2 votes]: A simpler way  is to use one of the consequences of the Lagrange's theorem applied on $\ln(x)$ function, namely:
$$\frac{1}{k+1} < \ln(k+1)-\ln(k)<\frac{1}{k} \space , \space k\in\mathbb{N} ,\space k>0$$
Then take $k=1,2,...,n$ values to the inequality and add them up. 
The proof is complete.