Convergence and limit of $a_{n+1} < \frac{a_n+a_{n-1}}{2}$

Question

Suppose $(a_n)$ is a sequence such that $\forall n: a_n \in (0,1)$ and $a_{n+1} < \frac{a_n+a_{n-1}}{2} (*)$.

(1) Is it true that $a_n$ converges?

(2) Find all possible limits of such sequences

Attempt

Let $k_{n+1}=\frac{a_{n+1}}{a_n}$. Then $(*)$ becomes $k_{n+1} < 0.5 + 0.5\frac{1}{k_n}$. So if $a_n$ was much bigger than $a_{n-1}$, then $a_{n+1}$ is smaller than $a_n$. Need to show that $\lim_{n\to\infty}k_n=1$ or $\lim_{n\to\infty} a_n = 0$. Got stuck

Edit

This question is a 100% duplicate of Convergence of sequence with $a_{n+2} < \frac{a_n + a_{n+1}}{2}$

Another similar (but not exactly same) question is If $2a_{n+2} \le a_{n+1}+a_n$, then $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$

Answer 1

HINT:

(1) Show that it is decreasing (for large $n$) and bounded.

(2) Is it obvious that every number from $[0,1)$ can be a limit?