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Let c_1, \ldots, c_n \in \mathbb{R} with n \geq 2 such that 0 \leq \sum^n_{i=1} c_i \leq n. Show that we can find integers k_1, \ldots, k_n such that \sum^n_{i=1} k_i = 0 and 1-n \leq c_i + n \cdot k_i \leq n for every i = 1, \ldots, n.

Another formulation:Let x_1, \ldots, x_n, with n \geq 2 be real numbers such that |x_1 + \ldots + x_n| \leq n. Show that there exist integers k_1, \ldots, k_n such that |k_1 + \ldots + k_n| = 0. and |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 for every i = 1, \ldots, n. In order to prove this, denote c_i = \frac{1+x_i}{2} for i = 1, \ldots, n, etc.

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