Question 39
The sequence $\alpha_0, \alpha_1, \alpha_2, \ldots, \alpha_\nu, \ldots$, $\nu \in \mathbb{N}$, with integer terms, not necessarily different, has the following properties:
(α) $0 \le \alpha_i \le i$, for every integer $i \ge 0$,
(β) $\binom{\kappa}{\alpha_0} + \binom{\kappa}{\alpha_1} + \dots + \binom{\kappa}{\alpha_\kappa} = 2^\kappa$, for every integer $\kappa \ge 0$.
Prove that for every integer $n \ge 0$, there exists $i \ge 0$ such that $\alpha_i = n$.
(α) $0 \le \alpha_i \le i$, for every integer $i \ge 0$,
(β) $\binom{\kappa}{\alpha_0} + \binom{\kappa}{\alpha_1} + \dots + \binom{\kappa}{\alpha_\kappa} = 2^\kappa$, for every integer $\kappa \ge 0$.
Prove that for every integer $n \ge 0$, there exists $i \ge 0$ such that $\alpha_i = n$.