After reading your comments, I come up with the following proof. I would like to know whether my proof is correct or not.

For weyl group $W$, each $w\in W$ can be expressed as $w=s_{\beta_l}\cdots s_{\beta_{2}} s_{\beta_1}$ for some distinct positive roots $\{\beta_1, \beta_2, \cdots, \beta_l\}\subseteq \Phi^+$.

**My proof:**

Let $w=s_{i_1}s_{i_2}\cdots s_{i_l}$ be a reduced expression with $s_{i_k}$ is the simple reflection of root $\alpha_{i_k}\in\Delta$.

Let $w_k=s_{i_1}s_{i_2}\cdots s_{i_k}$, $w_0=1$, $C_k=w_{k}C_+$ and $C_0=C_+$. By direct computation,
$
C_k=w_{k}C_+
=w_{k-1}s_{i_k}C_+
=w_{k-1}s_{i_k}w_{k-1}^{-1}w_{k-1}C_+
=w_{k-1}s_{i_k}w_{k-1}^{-1}C_{k-1}
=s_{\beta_k}C_{k-1},
$
where $\beta_k:=w_{k-1}\alpha_{i_k}=s_{i_1}s_{i_2}\cdots s_{i_{k-1}}(\alpha_{i_k})$ and $\beta_1=\alpha_{i_1}$.

Note that
$
wC_+
=C_l
=s_{\beta_l}C_{l-1}
=s_{\beta_l}s_{\beta_{l-1}}C_{l-2}=\cdots
=s_{\beta_l}s_{\beta_{l-1}}\cdots s_{\beta_2}C_1
=s_{\beta_l}s_{\beta_{l-1}}\cdots s_{\beta_1}C_+.
$
By the simply-transitivity of $W$, we get $w=s_{\beta_l}s_{\beta_{l-1}}\cdots s_{\beta_1}$.

Since $s_\alpha=s_{-\alpha}$ for all $\alpha\in\Phi^+$, we may WLOG assume $\beta_k\in\Phi^+$.

Suppose $\beta_k=\beta_j$ for some $j<k$, then $s_{\beta_k}=s_{\beta_j}$.
Then
$
w_{k-1}s_{i_k}w_{k-1}^{-1}=w_{j-1}s_{i_j}w_{j-1}^{-1}.
$
$
s_{i_j}=(s_{i_j}\cdots s_{i_{k-1}})s_{i_k}(s_{i_j}\cdots s_{i_{k-1}})^{-1},
$
from which we obtain, upon right-multiplying by $s_{i_j}\cdots s_{i_k}$.
$
s_{i_{j+1}}\cdots s_{i_{k-1}}
=s_{i_{j}}\cdots s_{i_{k}}.
$
Then
$
w=s_{i_1}\cdots s_{i_l}=s_{i_1}\cdots\hat{s_{i_j}}\cdots\hat{s_{i_k}}\cdots s_{i_l},
$
a contradiction to the fact that $w=s_{i_1}s_{i_2}\cdots s_{i_l}$ is a reduced expression.
Therefore, the claim follows.