$\limsup$ of functions and sequential $\limsup$.

Question

Let $f:(0,+\infty) \to \mathbb{R}$, a continous and non negative function and $g:(0,1) \to \mathbb{R}$ a non negativa continuous function, if for all sequence $(t_n)$ such that $t_n\to +\infty$
$$\limsup_{n \to +\infty}f(t_n) \leq \int_0^1g(s)\limsup_{n\to \infty} f(t_ns)ds$$ can we conclude that $$\limsup_{t \to +\infty}f(t) \leq \int_0^1g(s)\limsup_{t\to +\infty} f(t)ds?$$ If no, can we add some more condition for this to happen? Sorry for the lack of data, but I didn't find any reference that dealt with upper limit of functions at infinity.