Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{X} $, $\mathcal {Y} $ be two uniformly-integrable families on the same probability space $\Omega $.
Show that the following familie is also uniformly integrable: $$ \mathcal{X} +\mathcal {Y} =\{X+Y : X\in \mathcal{X} , Y\in\mathcal {Y} \}. $$
A family $\{X_i\}$ is uniformly integrable iff
a) $\sup_i E|X_i| <\infty$ and
b) for every $\epsilon >0$ there exists $\delta >0$ such that $\int_E |X_i|dP <\epsilon$ whenever $P(E) <\delta$.
Now you only need $|a+b| \leq |a|+|b|$ to verify that these two properties are satisfied by the sum $\mathcal X +\mathcal Y $