Limsup Squeeze Theorem

Theorem:If $$|X_n|\leq y_n, \forall n\geq n_0$$ and $$\limsup{y_n}=0$$. Then $$\lim_{n\rightarrow\infty}X_n=0$$ Proof: Since $$|X_n|\geq0$$, $$y_n\geq 0$$. Therefore, $$0\leq\liminf{y_n} \leq\limsup{y_n}$$. So, $$\liminf{y_n}=\limsup{y_n}=0$$, by the squeeze theorem. Thus, $$\lim{y_n}=0$$, but $$0\leq |X_n|\leq y_n \Rightarrow \lim{|X_n|}=0$$ But, $$-|X_n|\leq X_n \leq |X_n| \Rightarrow \lim{X_n}=0$$ QED