First of all, I am sorry for the ''not clear title' for this question but I cannot find a better way to describe this seemingly very simple and standard inequality,

So.. I am reading a paper '**Two-dimensional Navier-Stokes Equation Driven by a space time white noise**' by Daprato and Debussche. And I came across an inequality regarding a probability measure.(It is in the proof of main theorem p.198) It seems very strandard but I cannot see why it is.

$$\mathbb{P}\left[ \sup_{t\in [0,T]} \left| u_N(t,u_0) \right|_{\mathcal{B}^\sigma_{p,\rho}} \geq M \right]\leq \sum_{k=0}^{[T/t^*_M]} \mathbb{P}\left[\sup_{t\in [kt^*_M,(k+1)t^*_M]} \left| u_N(t,u_0) \right|_{\mathcal{B}^\sigma_{p,\rho}}\geq M \right]$$

Here, $\mathcal{B}^\sigma_{p,\rho}$ is a Besov space and $\mathbb{P}$ is a probability measure.

I initially thought it is a typo and the $M$ in the right hand side should be replaced with $M/[T/t^*_M]$ but I realized that there is a possibility that I am missing something. So I wanted to hear from someone else.

I thank in advance for any help with this.