Zeno's Barber

Way too hard to understand. Here's a simpler explanation:::

Let $(a_n)_{n=1}^{n=\infty}$ be a sequence of numbers in the field $a_n \in \mathbb R$ such that it follows a recurrence relation with $a_{n+1} = \frac12 a_n$ and initial term of $a_1 = \frac12$ . This shows that we have a geometric progression. Because the common ratio is $r = \frac12$ with $|r| < 1$ , we can show that the sum converge by ratio test. Furthermore, we note that limit supremum of this sum is 1, thus the partial sum is always less than 1. Thus you can't achieve the value of 1 by a finite partial sum. Quod Erat Demonstrandum.