Prove the following statement for a collection of natural numbers $x_1, x_2, \ldots , x_n$ and the set $P =\{1, 2, . . . , n\} $.
$$\left((x_1 + x_2 + \cdots + x_n) > \frac{n(n+1)}{2}\right)\rightarrow (\exists i\in P\,.\,x_i>i)$$
How would one prove this questions? Is there a way to phrase this question into word form? I am not sure what this question even mean.
The condition on the left can be re-written as $$\sum_{i=1}^{n}(x_i-i)>0\implies \exists i\text{ such that }x_i>i.$$ So now suppose that $\forall i$ we had that $x_i\leq i$ then we would get that $$\sum_{i=1}^{n}(x_i-i)\leq 0$$ which contradicts our hypothesis and so the proof is done.