<s>[INST] Proving that this sequence is convergent Given $a_1 = 2$, and $ a_{n+1} = \frac{a_n+5}{4} $ for all $n > 1$ , 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?
 [/INST] Let $x \in \mathbb R$. If  $x \gt 5/3$ then we always have
$\tag 1    \frac{5}{3} \lt \frac{x + 5}{4} \lt x$
Since our sequence $a_n$ is strictly decreasing and always greater than $5/3$, it has a limit which we denote by $\alpha$.
For the sequence to converge, $(a_{n+1} - a_n)$ must converge to zero, and
$\quad a_{n+1} - a_n =  \frac{a_n + 5}{4} - a_n = \frac{-3 a_n + 5}{4} $
and applying the limit operation,
$\quad \lim_{n\to +\infty} \frac{-3 a_n + 5}{4} = \frac{-3 \alpha + 5}{4} = 0$
Solving we find the limit of the sequence is equal to $5/3$.
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