Problem 1

Let

x_1,x_2,\ldots,x_n

be positive real numbers and let

S=x_1+x_2+\cdots+x_n

. Prove that

(1+x_1)(1+x_2)\cdots(1+x_n)\le 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots+\dfrac{S^n}{n!}

Step 2 of 4: Compare each power of S with the corresponding symmetric sum

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S^k=(x_1+\cdots+x_n)^k\ge k!e_k\qquad(1\le k\le n)

Detailed analysis

Fix a subset of $k$ distinct indices. Its product $x_{i_1}\cdots x_{i_k}$ appears in $S^k$ once for every permutation of those indices, hence at least $k!$ times. All other terms are positive, so summing over the subsets gives $S^k\ge k!e_k$ .