Proving that this sequence is convergent Given $a_1 = 2$, and $ a_{n+1} = \frac{a_n+5}{4} $ for all $n &gt; 1$ , is this sequence convergent? Give a formal proof in either case (converges or diverges).

Attempt: I do think this converges, but cannot say for sure. 

$a_1 = 2$

$a_2 = \frac{7}{4}$

$a_3 = \frac{27}{16}$

$a_4 = \frac{107}{64}$

and so on. I can see that it is decreasing and seems to be bounded below by something. But I do not know how to present it formally. 

Any help?
The sequence is descendant and positive, so it's convergent.

Where does it converge to ? Supposedly $l$ is its limit.

$l=\frac{l+5}{4}$ gives $l=\frac{5}{3}$
Hint: Use induction to show it's decreasing. Since it's positive it's bounded from below, thus it converges. Say it converges $L \in \mathbb{R}$. Apply the limit on both sides of the recursive formula of the sequence to find the value of L