How to apply triangle inequality in a probability statement?

Hint: prove that $$ \{\omega\in\Omega;|\bar{Y}(\omega)+X_n(\omega)-\mu| \ge \delta \} \subseteq \{\omega\in\Omega;|\bar{Y}(\omega)-\mu| \ge \frac{\delta}{2}\} \cup \{\omega\in\Omega;|X_n(\omega)| \ge \frac{\delta}{2}\} $$

Solution: Let $\omega\in\{\omega\in\Omega;|\bar{Y}(\omega)+X_n(\omega)-\mu| \ge \delta \} $, then if $\omega\notin\{\omega\in\Omega;|X_n(\omega)| \ge \frac{\delta}{2}\}$ one has $$ |\bar{Y}(\omega)+X_n(\omega)-\mu| \ge \delta ,\quad |X_n(\omega)| < \frac{\delta}{2} $$ So using triangle inequality one has $$ |\bar{Y}(\omega)-\mu| \ge |\bar{Y}(\omega)+X_n(\omega)-\mu|- |X_n(\omega)| \ge \delta- \frac{\delta}{2}=\frac{\delta}{2}. $$ Thus $$\omega\in\{\omega\in\Omega;|\bar{Y}(\omega)-\mu| \ge \frac{\delta}{2}\}$$

Tags:

Inequality

Probability

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