I have to solve the following task by using the Baire category theorem:
Let be $f: [0,\infty] \rightarrow \mathbb{R}$ a continous function such that
$$\forall t \geq 0: \lim_{n \rightarrow \infty} f(nt) = 0 $$
I have to show that $f$ satifies $$\lim_{t \rightarrow \infty} f(t)=0 $$
In our lecture we had the following version of the Baire category theorem:
Let be $(M,d)$ a complete metric space and $O_n \subset M$ open and dense. Then $\cap_{n \in \mathbb{N} }O_n $ is dense in M.
I don't know how to use here the Baire category theorem especially how to create such open and dense subsets. Can someone help me please?
Thanks in advance
