Weyl's asymptotic law on the compact segment $[0,a]$ with Dirichlet boundary conditions

Question

In the Matt Stevenson's article (Weyl's law), we have : $$N(\lambda) = \# \{k \in \mathbb{N} : \lambda_k < \lambda \} = \# \left\{k \in \mathbb{N} : \left(\frac{k \pi}{a}\right)^2 < \lambda \right\} = \max \left\{k \in \mathbb{N} : k < \frac{a \sqrt{\lambda}}{\pi}\right\} \sim \frac{a \sqrt{\lambda}}{\pi}.$$ Could anyone is able to give me an explicit formula for $\max \left\{k \in \mathbb{N} : k < \frac{a \sqrt{\lambda}}{\pi}\right\}$? I struggle trying to find why the Weyl's law for a bounded line is true.