Sum of Larger Ideals is Larger

Theorem
Let $R$ be a commutative ring with unity.

Let $\mathfrak a_1, \mathfrak a_2, \mathfrak b_1, \mathfrak b_2$ be ideals of $R$ such that:
 * $\mathfrak a_1 \subseteq \mathfrak a_2$

and:
 * $\mathfrak b_1 \subseteq \mathfrak b_2$

Then their sums satisfy:
 * $\mathfrak a_1 + \mathfrak b_1 \subseteq \mathfrak a_2 + \mathfrak b_2$

Proof
Let $x \in \mathfrak a_1 + \mathfrak b_1$.

By, there are $a \in \mathfrak a_1$ and $a \in \mathfrak b_1$ such that:
 * $x = a + b$

Then:
 * $a \in \mathfrak a_1 \subseteq \mathfrak a_2$

and:
 * $b \in \mathfrak b_1 \subseteq \mathfrak b_2$

Thus, by :
 * $x \in \mathfrak a_2 + \mathfrak b_2$