calculus - A Proof for a Sequence Convergence

Here's my question to prove:

Define $a_n$ to be a sequence such that:

$$a_1=\frac{3}{2}$$
$$a_{n+1}=3-\frac{2}{a_n}$$

Prove that $a_n$ is convergent and calculate its limit.

Solution

Prove by induction that $a_n$ is monotonic increasing:

• For $n=2$, $a_2=3-\frac{4}{3}=\frac{5}{3}>\frac{1}{2}$


• Assume that $a_n>a_{n-1}$



• For $n=k+1$:

  $$3-\frac{2}{a_n} - (3-\frac{2}{a_{n-1}})=-\frac{2}{a_n}+\frac{2}{a_{n-1}}>0$$ , since $a_{n-1}\frac{2}{a_n}$




• Therefore, the sequence is monotonic increasing.

Prove that $2$ is an upper bound of the sequence. Therefore, it is monotonic increasing and bounded, thus convergent (induction).

Now I think that $\lim_\limits{n\to\infty} a_n=2$, and I want to prove it with the squeeze theorem.

Is my solution, correct?

Is there a way to find supremum here?

Thanks,

Alan