Incorrect mathematical integration

I once explained to a speaker that additions of relativistic speeds were not done in a common way, and that for example 0.5c+0.5c did not make c.
This Internet user refused to believe me.
For what? Because it is very difficult to give water to a donkey who is not thirsty, and who categorically refuses to understand or discuss.
I think that this makes most of the speakers smile, because they know A LITTLE realtivity, and if they do not necessarily know the general formula for adding relativistic speeds, they at least know the longitudinal formula that is w=(v +u)/(1+v.u/c²) or here w=0.8c.
But we must go further. Physicists don't make this kind of mistake, but they do make others. I told Paul B. Andersen that his magnificent integration formula
<http://news2.nemoweb.net/jntp?EKV4LWfwyF4mvRIpW8X1iiirzQk@jntp/Data.Media:1>
was incorrect PHYSICALLY even though mathematically it was obviously perfect.
Paul doesn't want to believe me. This confuses him.
However he is wrong and I pointed out to him that if we could integrate all the proper times, to obtain the sum of the total proper time, we could not do it with improper times, the sum of which segment by segment was greater than the total evolution.
A bit like realtivist speed additions where the sum is not equal to the common, mathematical sum.
Paul doesn't want to believe me, because he wasn't taught that way, and he complains about me.
Why doesn't he complain about those who taught him incorrectly?
R.H.