How do you prove that the Cesàro mean of the harmonic series diverges.
I know that the harmonic serie
$$\sum_{k=1}^{\infty} \frac{1}{k}$$ diverges,
However, I'm having trouble proving that it also diverges in the sense of Cesaro.
I have that
$$\sigma_n = \frac{1}{n}\sum_{m =1}^n \sum_{k =1}^m \frac{1}{k}$$
So, we can write
$$\sigma_n = \frac{1}{n}\sum_{k=1}^n \frac{1}{k}\sum_{m=k}^{n}1$$ and then we have $$\sigma_n = \frac{1}{n}\sum_{k=1}^n \frac{1}{k}(n+1-k)$$ but then I dont know how to continue. I thought that maybe the $\limsup_{n\geq k} \sigma(n)$ could maybe help but i'm really stuck and would appreciate any help!