If $f(x)$ is convex. Then, is $\sum_{a \in A}f(x).\alpha_a + \sum_{b \in B}f(x).\beta_b$ also convex

Question

If $f(x)$ is convex. Then:

1. Is $\sum_{a \in A}f(x).\alpha_a$ also convex?
2. If the answer to first question is yes and $\sum_{a \in A}f(x).\alpha_a$ is convex, is $\sum_{a \in A}f(x).\alpha_a + \sum_{b \in B}f(x).\beta_b$ also convex?

How can I prove them as a lemma?

$f(x)$ is a convex non-linear function and $\alpha$ and $\beta$ are constant variables.