Learning from a miss

I am learning to throw a ball into a basket. My first throw misses by a hundred centimetres, far too much to the right.

First throw: error $e_0=100$ cm.

$e_0=100$

I do not give up, and I do not throw the same way again. I look at the miss and fix part of it: after each throw, seventy in a hundred of the earlier error remains. So the error goes 100, then 70, then 49, then 34.3.

After each throw seventy per cent of the error remains: $e_{n+1}=0.7e_n$, so $e_n=100\cdot 0.7^n$. That gives $e_1=70$, $e_2=49$ and $e_3=34.3$.

$e_{n+1}=0.7e_n\Rightarrow e_n=100\cdot 0.7^n$

The basket accepts any throw whose error is under twenty centimetres. The fifth throw is off by 24.01 and stays outside, but the sixth is off by 16.807 and drops in.

The basket accepts errors under $20$ cm. $e_4=24.01>20$ but $e_5=16.807<20$, so the sixth throw drops in. So $\exists n,e_n<20$, and the smallest such $n$ is $5$.

$e_4=24.01>20>16.807=e_5$, $\exists n,e_n<20$

If I change nothing, the error stays 100 for ever and the ball never goes in. An error that I use shrinks, and comes closer and closer to zero. Someone who never missed would not know how to correct.

If I change nothing, then for every $n$ we have $e_n=100$. If I use the error, ${\lim }_{n\to \infty }{e}_n=0$.

$\forall n,e_n=100$; ${\lim }_{n\to \infty }{e}_n=0$