Suppose that
![x_1, x_2, x_3, \ldots](/media/m/e/7/e/e7eab3322da37c596b1e9d6c58193bdd.png)
are positive real numbers for which
![x^n_n = \sum^{n-1}_{j=0} x^j_n](/media/m/5/d/3/5d3ca77c48ff3d171daeb288fce5fb61.png)
for
![n = 1, 2, 3, \ldots](/media/m/d/5/9/d59e410f8c37653f899a4e74178138f0.png)
Prove that
%V0
Suppose that $x_1, x_2, x_3, \ldots$ are positive real numbers for which $$x^n_n = \sum^{n-1}_{j=0} x^j_n$$ for $n = 1, 2, 3, \ldots$ Prove that $\forall n,$ $$2 - \frac{1}{2^{n-1}} \leq x_n < 2 - \frac{1}{2^n}.$$