Suppose that $$\lim_{n \to \infty} f(n\lambda ) = 0 \quad \forall \lambda \in (0,1).$$
Is it necessary that $$\lim_{x \to +\infty} f(x) = 0?$$
I tried to construct an example to show it's not true, but I failed. Can you help me?
Suppose that $$\lim_{n \to \infty} f(n\lambda ) = 0 \quad \forall \lambda \in (0,1).$$
Is it necessary that $$\lim_{x \to +\infty} f(x) = 0?$$
I tried to construct an example to show it's not true, but I failed. Can you help me?
The usual question along these lines assumes that $f$ is continuous. Then Arpit's hint is helpful.
But for this one... Define $f(x) = 1$ if $x=\pi^m$ for some $m \in \mathbb N$ and $0$ otherwise. For each $\lambda$, the sequence $(f(n\lambda))_{n \in \mathbb N}$ is zero except for at most one term, so converges to zero. But $f(x)$ does not converge to zero.