If found the following use of inequalities in a proof for the Squeeze Theorem:
$$\vert a_n - l \vert \lt \epsilon \iff - \epsilon \lt a_n - l \lt \epsilon$$
In deciding, why this is true I came up with the following proof:
Proposition: The following statements are equivalent, that is, they are either both true or both false:
$$\vert a - b \vert \lt c$$
$$-c \lt a - b \lt c$$
Proof: Suppose that $\vert a - b \vert \lt c$. Since $a-b$ can be either positive or negative, we have
$$\vert a - b \vert = a - b \lt c \tag{1}\label{1}$$
or
$$\vert a - b \vert = -(a-b) \lt c \tag{2}\label{2}$$.
From $\eqref{2}$ it follows that $-c \lt a - b$ and combining this with $\eqref{1}$ we have
$$-c \lt a - b \lt c$$
The converse follows easily by doing those steps backwards. $\blacksquare$
Is this correct?