Proving that $(a_n) $ defined by induction to be: > $a_1=2.2$, $a_{n+1}=5-\frac6{a_n}$ is converging and finding the limit 
Let there be the sequence $(a_n) $ defined by induction to be:
  $a_1=2.2$, $a_{n+1}=5-\frac6{a_n}$ $\forall n\ge 1$
Prove that the sequence is converging and calculate it's limit.

So what needs to be done is to show that it's increasing and bounded and to find the limit. 
Finding the limit we know that $a_n\to \alpha $ which leads to $ \alpha=5-\frac6{\alpha}$ after solving the quadratic equation we get: $ \alpha_{1,2}=-1,6$ so we know that the limit is 6 because it can't be negative. 
Increasing, by induction, we need to show that $a_n\le a_{n+1}$ so we get to: $a_{k+2}= 5-\dfrac6{a_{k+1}} \ge 5-\dfrac6{a_k}=a_{k+1}\to...\to a_{k}\le a_{k+1}$.
Is my way correct ? how do I show that it's bounded ? anything missing ?
NOTE: I need to know how to solve this rigoursly as I study for a test so please let me know of anything that needs to be mentioned in the test. 

Hint: 
$2.2\leq a_{n}\leq3$ implies $2.2\leq a_{n+1}\leq3$. 
You can show that under these conditions $a_{n+1}>a_n$
The equation $$\alpha=5-\dfrac{6}{\alpha}$$ has $\alpha=2$ and $\alpha=3$ as solutions.
