Why does a positive definite matrix defines a convex cone?

Ok. Let $P$ be the set of all positive definite matrix. Im gonna show that if $X,Y\in P$ and $\alpha,\beta>0$ then $\alpha X+\beta Y\in P$. Note \begin{eqnarray} x^{\top}(\alpha X+\beta Y)x &=& \alpha x^{\top} Xx+\beta x^{\top} Yx \nonumber \\ &>& 0\end{eqnarray}

Above, i have used algebraic properties of matrix product and the positive definitiness of $X$ and $Y$. With this you can conclude that $\alpha X+\beta Y\in P$