$0Let $a_n$ be a sequence of positive real numbers such that
$$a_n<\frac{a_{n-1}+a_{n-2}}{2}$$
Show that $a_n$ converges.

It is clear that $(a_n)_{n\in\mathbb{N}}$ is bounded between $\ \max \{a_1,a_2\} $ and $0.$ Therefore by the Bolzano-Weierstrass theorem, $\ (a_n)_{n\in\mathbb{N}}\ $ has at least one limit (accumulation) point in the interval$\ [\ 0,\ \max \{a_1,a_2\}\ ].$
Now suppose, by way of contradiction, that $\ (a_n)\ $ has at least two limit points, $\ l_1\ $ and $\ l_2,\ $ with $\ l_1<l_2.\ $
Lemma: If  $\ a_{n+1} > a_n,\ $ then $\ a_k < a_n + \frac{3}{4}\left( a_{n+1} -a_n\right)\quad \forall\ k\geq n+2.$
Next, since $\ l_1\ $ and $\ l_2\ $ are limit points, $\ \exists\ N_1\ $ such that, $\ a_{N_1} < \frac{l_1+l_2}{2},\ $ and $\ a_{N_1 + 1} > \frac{l_1+l_2}{2}.\ $ The Lemma implies then that:
$$\ a_k < a_{N_1} + \frac{3}{4}\left( a_{N_1+1} -a_{N_1}\right) \leq \frac{l_1+l_2}{2} + \frac{3}{4}\left( a_{N_1+1} - \frac{l_1+l_2}{2}\right)$$
$$\implies a_k - \frac{l_1+l_2}{2} < \frac{3}{4}\left( a_{N_1+1} - \frac{l_1+l_2}{2}\right)  \quad \forall\ k\geq N_1+2.\quad (1) $$
Continuing, $\ \exists\ N_2>N_1+2\ $ such that, $\ a_{N_2} < \frac{l_1+l_2}{2},\ $ and $\ a_{N_2 + 1} > \frac{l_1+l_2}{2}.\ $ Since $\ N_2 > N_1 + 2,\ $ all $\ k\geq N_2\ $ satisfies $\ (1)\ $ also. Thus,
$$ a_k - \frac{l_1+l_2}{2} < \frac{3}{4}\left( a_{N_2+1} - \frac{l_1+l_2}{2}\right) < \frac{3}{4}\left( \frac{3}{4} \left( a_{N_1+1} - \frac{l_1+l_2}{2}\right) \right) = \left( \frac{3}{4} \right)^2 \left( a_{N_1+1} - \frac{l_1+l_2}{2}\right) $$
$$ \quad \forall\ k\geq N_2+2.\quad (2) $$
Continuing in this way, we can find and increasing sequence of integers $\ (n_j)_{j\in\mathbb{N}}\ $ such that for each $\ j,\ $ we have:
$$ a_k \leq \frac{l_1+l_2}{2} + \left(\frac{3}{4}\right)^j \left( a_{N_1 + 1} - \frac{l_1+l_2}{2} \right)\quad \forall\ k\geq N_j + 2. $$
But then, since $\  \left(\frac{3}{4}\right)^j \to 0,\ $ as $\ j\to\infty,\ $ it follows that $\ \displaystyle\limsup_k\ a_k\ \leq \frac{l_1+l_2}{2} $ contradicting the fact that $\ l_2\ $ is a limit point of $\ (a_n)_n.$
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I have left the proof of the lemma, and a couple of other details as an exercise to the reader, but these things should be relatively straightforward.
