Checking whether sequence $ x_n = \ln(n^2 + 1) - \ln(n) $ converges or diverges

Question

I have to show whether

$$ x_n = \ln(n^2 + 1) - \ln(n) $$

converges or diverges.

I can write

$$ x_n = \ln(n^2 + 1) - \ln(n) = \ln\left(\frac{n^2+1}{n}\right) = \ln\left(n + \frac{1}{n}\right). $$

I know that $n+\frac{1}{n} \to \infty$ for $n \to \infty$.

Since $\ln(x) \to \infty$ for $x \to \infty$, the series $x_n = \ln(n^2 + 1)-\ln(n)$ diverges.

Is this correct? I think it is too easy. Have I forgot something? Can I use some lemma?