Prove : $C_{k}(x)+C_{k+1}(x)\geqslant 1$ , $x_{k}\leqslant x\leqslant x_{k+1}$ Suppose $-\infty < x_{1}< x_{2}< \cdots < x_{n}< +\infty (n\geqslant 2)$ . And suppose a algebraic polynomial $C_{k}(x)$ $(k=1,2,\cdots ,n)$  ($degree\leqslant n-1$ ) satisfy :
$$C_{k}(x_{i})=\left\{\begin{matrix} 0, & i\neq k,\\ 1, & i=k, \end{matrix}\right. (i=1,2,\cdots ,n)$$ 

Prove : $C_{k}(x)+C_{k+1}(x)\geqslant 1$ , $x_{k}\leqslant x\leqslant x_{k+1}$ $(1\leqslant k\leqslant n-1)$ .  

Let $C_{k}(x_{i})=a_{k}(x-x_{1})(x-x_{2})\cdots (x-x_{k-1})(x-x_{k+1})\cdots (x-x_{n})$ , $(k=1,2,\cdots ,n)$ .
$a_{k}=\frac{1}{(x-x_{1})(x-x_{2})\cdots (x-x_{k-1})(x-x_{k+1})\cdots (x-x_{n})} $ .
It satisfies the condition . But I can't prove the inequality .
Hope someone helps me . 
Appriciate!

Hint, rather than a complete answer. Try to prove by contradiction. Sketch some examples with 4-5 points to see where the problem is.
1) $P = C_k + C_{k+1}$ is polynomial of degree $\leq n-1$
2) $P$ has $n-2$ zeros at known locations
3) How many zeros does $P^{\prime}$ have?
