A sequence $(x_n)_{n\in\mathbb N}$ is defined by $$x_0=0, \qquad x_{n+1}=\frac23(x_n+1)\text{ for }n=1,2,\dots$$ Prove that this sequence converges and find the limit.
The infimum of $(x_n)$ here is obviously 0, however, I do not know how to find the supremum and the limit of $(x_n)$.
On
If $L$ denotes the limit value, we should have $L = \frac23(L+1)$ or $L=2$. Then writing $y_n = 2- x_n$, we want to show that $y_n \to 0$ and our recursion becomes $y_{n+1} = \frac23 y_n$, with starting value $y_0 = 2$.
On
If you can show that $$x_{n-1}\le x_n \qquad\Rightarrow\qquad \frac23(x_{n-1}+1)\le \frac23(x_n+1),$$ you should be able to see that this sequence is monotone.
If you can show that $$x_n\le 2 \qquad\Rightarrow\qquad \frac23(x_n+1)\le2,$$ you should be able to show that $x_n\le 2$. The inequality $x_n\ge0$ is easy, so this sequence is bounded.
Every bounded monotone sequence is convergent.
From the other answers you already know how to find the limit.
On
Make a guess of the limit and then use the monotone convergence theorem. You have to show that the sequence is bounded and monotonic (i.e. increasing) and then conclude by MCT that it converges.
To guess the limit solve $L=\frac{2}{3}(L+1)$.
Edit: The guess of the limit is used in the MCT part. Show that $x_n \leq 2 \forall n \in \mathbb{N}$
On
If a recurrence relation has a limit, then it is a fixed point of the recurrence. So the only possible limit of
$$ x_{n+1}=\frac23(x_n+1)$$
... is $x$ such that
$$x =\frac23(x+1)$$
i.e. $$x = 2$$
... independent of the initial value $x_0$.
If we now define
$$ y_n = x_n - 2 \text{ for }n=1,2,\dots$$
then
$$ y_{n+1} + 2 = \frac23(y_n+3)$$
so
$$ y_{n+1} = \frac23y_n$$
... which clearly converges to $0$ for any $y_0$.
On
$3x_{n+1}-3x_n=2x_n+2-2x_{n-1}-2\implies 3x_{n+1}-5x_n+2x_{n-1}=0$
Therefore the characteristic equation is,
$$3t^2-5t+2=0$$
$\therefore t=\dfrac{2}{3}, 1$
$${x_n=a+b\left(\dfrac{2}{3}\right)^n}\tag{1}$$
To find $a$ and $b$ we note that,
$x_0=0=a+b\tag{2}$ $x_1=\dfrac{2}{3}=a+b\left(\dfrac{2}{3}\right)\tag{3}$
Solving we get $a=2,b=-2$.
Thus $${x_n=2\left(1-\left(\dfrac{2}{3}\right)^n\right)}$$
Therefore,
$$\color{blue}{\boxed{\displaystyle\lim_{n\to\infty}x_n=2}}$$
I have intentionally skipped the proof of existence of $\displaystyle\lim_{n\to\infty}x_n$. Why can we assume that the limit indeed exists?
To guess the limit in this sort of question, note that, if the limit is $L$, then both $x_n$ and $x_{n+1}$ tend to $L$. So replace them both by $L$ in the recurrence relation and solve for $L$. Then you just have to show that the sequence converges to this limit.