A sequence was defined by a equation The sequence was defined by the equations:
$${a}_{1}=a\left(\in R \right),{a}_{n+1}=\frac{2{{a}_{n}}^{3}}{1+{{a}_{n}}^{4}},n\geq 1.$$
show that
$\left(a \right)$The given sequence is convergent. (whatever $a\in \mathbf{R}$)
$\left(b \right)$
Finding its all possible limit.
$\left(c \right)$
Dividing $\mathbf{R}$ into several intervals,such that if the initial value ${a}_{1}$ lies in the same interval,then sequence $\begin{Bmatrix}
{a}_{n}
\end{Bmatrix}$ has the same limit.

Hint: Finding all possible possible limits for part b) is very simple. Suppose one possible limit of the sequence is $L=\lim_n a_n$. Taking the limits of both sides of the recurrence relation you find:
$$L=\frac{2L^3}{1+L^4}.$$
That is, if any limits exist then they must satisfy the above equation.
*Note: just because a real number satisfies the above equation doesn't guarantee there actually does exist a sequence converging to that limit, so be careful not to over think part c).
