We have to show:
Let $\theta_0$ be a k-dim vector. Show, that the following statements are equivalent:
(1) $\hat{\theta}_n$ is consistent for $\theta_0$.
(2) For each component $i= 1,...,k$: $\hat{\theta}_{n,i}$, is $\hat{\theta}_{n,i}$ consistent for $\theta_{0,i}$, where $\hat{\theta}_{n,i}$ and $\theta_{0,i}$ are the i'th component of the respective vector.
Hint: $||x||_\infty \leq ||x||_2 \leq \sqrt{k}||x||_\infty$ as well as: $P(A+B \geq \varepsilon) \leq P(A\geq \frac{\varepsilon}{2}) + P(B\geq \frac{\varepsilon}{2})$
Any suggestions would be greatly appreciated.
So an estimator is consistent if:
$\lim_{n \rightarrow \infty} P(|\hat{\theta}_n - \theta| > \varepsilon) = 0$, however using the inequality: $P(A+B \geq \varepsilon) \leq P(A\geq \frac{\varepsilon}{2}) + P(B\geq \frac{\varepsilon}{2})$ doesn't make much sense here (from my understanding).