Let $x_1,x_2,x_3...$ be all consecutive positive roots of the equation $\tan x = x.$ Find $\lim_{n \rightarrow \infty}(x_n-x_{n-1}).$

Question

Let $x_1,x_2,x_3...$ be all consecutive positive roots of the equation $\tan x = x.$ Find $\lim_{n \rightarrow \infty}(x_n-x_{n-1}).$

I feel like the answer is $0$, as the gap between $x_n$ and $x_{n-1}$ is getting smaller and smaller.

My attempt: Note that $ n\pi\leq x_n \leq (n+1)\pi$. However, $\pi \leq x_n - x_{n-1} \leq \pi$ and I couldn't use Squeeze theorem to conclude.

Any hint would be appreciated.