BALTIC-WAY 2013 Problem 6

Santa Claus has at least $n$ gifts for $n$ children. For $i \in\{1,2, \ldots, n\}$, the $i$-th child considers $x_{i}>0$ of these items to be desirable. Assume that

$$\frac{1}{x_{1}}+\ldots+\frac{1}{x_{n}} \leq 1$$

Prove that Santa Claus can give each child a gift that this child likes.