ALIBABA 2021 Problem 6

Let $M(t)$ be measurable and locally bounded function, that is, $$M(t) \le C_{a,b}, \forall 0 \le a \le t \le b<\infty$$ with some constant $C_{a,b}$ , from $[0,\infty)$ to $[0,\infty)$ such that $$M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \forall t \ge 0.$$ Show that $$M(t) \le 10+2\sqrt{5}, \forall t \ge 0.$$