How to prove $\ln N\ge 1/2+1/3+1/4+\cdots+1/N$ Let $\ln(x)=\int_1^x\frac{1}{t}dt$. I want to show that $\ln N\ge 1/2+1/3+1/4+\cdots+1/N$. The hint is Riemann sum, but I can only get $\int_1^N\frac{1}{t}dt&gt;1\cdot\ln 1+1\cdot\ln2+\cdots+1\cdot\ln N$ (use the fact that integral $\ge$ lower sum). How to prove that? Or is my attempt wrong?

And also for $2^k$, how to show that $1/2+1/3+\cdots+1/N\ge k/2$?
Hint: If $f(t)$ is decreasing, 
$\int_n^{n+1} f(x)\; dx \ge f(n+1)$.  Add these for $n$ from $1$ to $N-1$.
for some function $f(x)$:
$$\int_a^bf(x)dx\ge\sum_{x=a}^bf(x)$$
so we know that:
$$\int_1^N\frac1xdx\ge\sum_{i=1}^N\frac1i$$
and we know that:
$$\int_1^N\frac1xdx=\ln(N)$$