Sum of Positive Definite Matrices – Positive Definite?

Question

I have two matrices, which are square, symmetric, and positive definite.
I would like to prove that the sum of the two matrices still have the same properties,
that is square, symmetric, and positive definite.
The first two properties are obvious, what about the positive definite property.
Any clue to the proof?
Thank you.

Answer 1

A real matrix $M$ is positive-definite if and only if it is symmetric and $u^TMu > 0$ for all nonzero vectors $u$.

Now if $A$ and $B$ are positive-definite, $u^T(A+B)u = \ldots?$