Proving a set is bounded. Let  $E = \{x \in\mathbb R^p : \sum_{i = p}^p X_i^2/\alpha_i^2 \leq 1\}$
Prove that $E$ is closed and bounded. 
To prove that $E$ is closed I used the fact that the boundary of the set $E$ is equal to $\{x \in\mathbb R^p : \sum_{i = p}^p X_i^2/\alpha_i^2 = 1\}$ and the boundary is contained in the $E.$ So $E$ is closed.
However I do not know how to prove the fact that it is bounded.

Try to get a lower bound on $\sum\frac{x_i^2}{\alpha_i^2}$ involving the Euclidean norm of $x$. For example:
$$\sum\frac{x_i^2}{\alpha_i^2}\geq\min_i\left(\frac1{\alpha_i^2}\right)||x||_2^2$$
