Answer
The sequence diverges. See the explanation below.
Work Step by Step
We have
$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{2x,}&{0 \le x \lt 0.5}\\
{2x - 1,}&{0.5 \le x \lt 1}
\end{array}} \right.$
We evaluate the sequence and list them in the following table:
$\begin{array}{*{20}{c}}
{f\left( x \right)}&{{\rm{equal to}}}\\
{f\left( {0.2} \right) = }&{0.4}\\
{f\left( {f\left( {0.2} \right)} \right) = }&{0.8}\\
{f\left( {f\left( {f\left( {0.2} \right)} \right)} \right) = }&{0.6}\\
{f\left( {f\left( {f\left( {f\left( {0.2} \right)} \right)} \right)} \right) = }&{0.2}\\
{f\left( {f\left( {f\left( {f\left( {f\left( {0.2} \right)} \right)} \right)} \right)} \right) = }&{0.4}\\
{f\left( {f\left( {f\left( {f\left( {f\left( {f\left( {0.2} \right)} \right)} \right)} \right)} \right)} \right) = }&{0.8}\\
{f\left( {f\left( {f\left( {f\left( {f\left( {f\left( {f\left( {0.2} \right)} \right)} \right)} \right)} \right)} \right)} \right) = }&{0.6}\\
{f\left( {f\left( {f\left( {f\left( {f\left( {f\left( {f\left( {f\left( {0.2} \right)} \right)} \right)} \right)} \right)} \right)} \right)} \right)... = }&{...}
\end{array}$
In the table above, we see that the sequence keeps repeating the set of values $0.4,0.8,0.6,0.2$. Therefore, we conclude that the sequence does not converge to some finite limit. Hence, it diverges.
