Convergence of weighted regression solution to the solution of L1 regression

I have read in this paper that the weighted regression solution converges to the solution of L1 regression,

for weights, $$W_i=1/|y_i-\hat y_i|$$

I worked this out but unfortunately, I lost.

Could anyone provide me with a reference that includes this derivation?

minimize the sum of squared errors $$\sum_i \epsilon_i(\theta)^2$$ with $$\epsilon_i(\theta) = y_i - \mathbf x_i^\top \theta$$ which is simply ordinary least squares estimation for $$\theta$$. But clearly $$\sum_i \epsilon_i(\theta)^2 = \sum_i \lvert\epsilon_i(\theta)\lvert^2$$
now use weights and observe that $$\lvert y_i - \hat y_i(\theta)\lvert = \lvert \epsilon_i(\theta)\lvert$$ then with weights $$w_i=1/\lvert y_i - \hat y_i(\theta)\lvert$$ it follows that $$\sum_i \epsilon_i(\theta)^2w_i = \sum_i \lvert\epsilon_i(\theta)\lvert^2w_i= \sum_i \lvert\epsilon_i(\theta)\lvert^2/\lvert y_i - \hat y_i(\theta)\lvert= \sum_i \lvert\epsilon_i(\theta)\lvert^2/\lvert \epsilon(\theta)\lvert =\sum_i \lvert\epsilon_i(\theta)\lvert$$ which is LAD.