# Proof of convergence of a sum of mean-consistent estimators

After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these exercises.

Suppose $\theta_n$ and $\phi_n$ are mean consistent estimators for $\theta,\phi$.

Prove: $\theta_n +\phi_n\rightarrow^{\mathcal{L}_1}\theta +\phi$ and $\max(\theta_n,\phi_n)\rightarrow^{\mathcal{L}_1}\max(\theta,\phi)$

Thus far I have gathered/surmised that the definition of mean consistent estimators essentially says:

$\theta_n\rightarrow^{\mathcal{L}_1}\theta$

$\phi_n\rightarrow^{\mathcal{L}_1}\phi$

Even under the assumption that $\theta_n\rightarrow\theta$ a.s, I am somewhat confused on how to proceed.

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For the first, it's a consequence of the triangular inequality. For the sequence, we have $\max\{a,b\}=\frac{a+b+|a-b|}2$, then use inverse triangular inequality. – Davide Giraudo Jul 28 '12 at 15:58

Integrate the inequalities $$|(\theta_n+\phi_n)-(\theta+\phi)|\leqslant|\theta_n-\theta|+|\phi_n-\phi|,$$ and $$|\max\{\theta_n,\phi_n\}-\max\{\theta,\phi\}|\leqslant|\theta_n-\theta|+|\phi_n-\phi|.$$