Is there a proof that each Cauchy sequence converges in more constructive flavor, without using Bolzano-Weistrass? Is this valid?
For each $k$ there exist $h_k$ $$|x_n-x_m|<2^{-k}, \space \space \space \space \space \space \space \space \space n,m>h_k$$
Then for $n,m>h_k$ $$x_m-2^{-k}\leq x_n\leq x_m+2^{-k}$$
so that for all $n \geq h_k$ $$\text{inf}_m \{x_m-2^{-k}:m\geq h_k \} \leq x_n\leq \text{sup}_m \{x_m+2^{-k}:m\geq h_k \}$$
Now define $r'=\text{sup}_k \text{inf}_m \{x_m-2^{-k}:m\geq h_k \} $, and $r= \text{inf}_k \text{sup}_m\{x_m+2^{-k}:m\geq h_k \}$. Note that since Cauchy sequences are bounded the definitions give valid real numbers.
Now we note that $r=r'$ as $$|r-r'|\leq 2^{-k+1} \rightarrow 0$$
And therefore $|x_k-r|\leq 2^{-k+1}\rightarrow 0$, which proves the theorem.