Solve {l}{||a_{n}|-|a||<|a|/2}{Rightarrow-|a|/2+|a|<|a_{n}|<|a|/2+|a|}{Rightarrowquad|a|/2<|a_{n}|<frac{3|a|}{2}}

$$\left. \begin{array}{l}{ | | a _ { n } | - | a | | < \frac { | a | } { 2 } }\\{ \Rightarrow - \frac { | a | } { 2 } + | a | < | a _ { n } | < \frac { | a | } { 2 } + | a | }\\{ \Rightarrow \quad \frac { | a | } { 2 } < | a _ { n } | < \frac { 3 | a | } { 2 } }\end{array} \right.$$

Evaluate (complex solution)

$$||a_{n}|-|a||<\frac{|a|}{2},\ \frac{|a|}{2}<|a_{n}|$$

Solve for a_n

$$a_{n}\in \left(-\frac{3|a|}{2},-\frac{|a|}{2}\right)\cup \left(\frac{|a|}{2},\frac{3|a|}{2}\right)$$

$$a\neq 0$$

Solve for a

$$\left\{\begin{matrix}a\in (|a_{n}|,2|a_{n}|)\cup (-2|a_{n}|,-|a_{n}|)\text{, }&a_{n}\neq 0\\a\in \mathrm{R}\setminus ((-\infty,-\frac{2a_{n}}{3}]\text{, }a_{n}\geq 0\cup (-a_{n},\infty)\cup [\frac{2a_{n}}{3},-\frac{2a_{n}}{3}]\text{, }a_{n}\leq 0\cup [2|a_{n}|,\infty)\cup (-\infty,a_{n})\cup (-a_{n},a_{n})\text{, }a_{n}>0\cup (-\infty,-2|a_{n}|])\text{, }&a_{n}<0\\a\in \mathrm{R}\setminus ((-\infty,\frac{2a_{n}}{3}]\text{, }a_{n}\leq 0\cup (a_{n},\infty)\cup [-\frac{2a_{n}}{3},\frac{2a_{n}}{3}]\text{, }a_{n}\geq 0\cup [2|a_{n}|,\infty)\cup (-\infty,-a_{n})\cup (a_{n},-a_{n})\text{, }a_{n}<0\cup (-\infty,-2|a_{n}|])\text{, }&a_{n}>0\end{matrix}\right.$$