<s>[INST] show that the following recursive series is bounded and monotonic I need to show that the following recursive series $$a_1=1, \quad a_{n+1}=\frac{a_n+3}{5}$$ is bounded and monotonic.
At first, I want to show that the series is bounded from below with $\frac{3}{4}$. By induction:
For $n=1$ we have $a_1=1>\frac{3}{4}$. We assume it is true for $n$, i.e. $a_n>\frac{3}{4}$ and we check for $a_{n+1}$:
$$a_{n+1}=\frac{a_n+3}{5}>\frac{\frac{3}{4}+3}{5}=15>\frac{3}{4}.$$
We conclude that $a_n>\frac{3}{4}$ for all n.
The series is increasing because
$a_{n+1}-a_{n}=\frac{-4a_n+3}{5}=\frac{-4(a_n-\frac{3}{4})}{5}<0$$
so we have also $a_n \leq a_1=1$.
Is my solution ok? Thank you for your help.
 [/INST] $$
\begin{align}
a_{n+1}-\frac34
&=\frac{a_n+3}5-\frac34\\
&=\frac{a_n-\frac34}5
\end{align}
$$
Thus, if $a_n\ge\frac34$, then $a_{n+1}\ge\frac34$. Therefore, since $a_1=1$, $a_n\ge\frac34$.
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