Limsup Squeeze Theorem

Theorem:
If $$|X_n|\leq y_n,

\forall n\geq n_0$$ and $$\limsup{y_n}=0$$,

then $$\lim_{n\rightarrow\infty}X_n=0$$

Direct Proof:
Since $$|X_n| \geq 0$$, $$y_n\geq 0$$.

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 $$0\leq |X_n|\leq y_n \Rightarrow \lim{|X_n|}=0$$

Therefore $$\lim{-|X_n|}=0$$

Then since $$-|X_n|\leq X_n \leq |X_n| \Rightarrow \lim{X_n}=0$$ by the squeeze theorem.

 Q.E.D.