Convergence of sequence of stochastic processes

Question

I want to show that for any $y_0>0$ $$\sup_{0<y<y_0} y^{-1} X_n(y) \overset{P}{\to}0, \hspace{25mm}(1)$$ Here $X_n(y)$ is to be regarded as a sequence of real-valued stochastic process defined on $[0, \infty]$.

The situation is as follows, I can show that for any $0< \epsilon <y_0$ $$\sup_{\epsilon<y<y_0} y^{-1} X_n(y) \overset{P}{\to}0$$ Also I can show that $$\lim_{y \to 0}y^{-1}X_n(y)=0 \text{ a.s.}$$ Is there any way to connect these two so that we get the first statement (1)?

Answer 1

$X_n(y)=\frac {\sin (n y)} {\sqrt {ny}}$ is a counterexample.