Prove minimum of $\sum_{i=1}^n=S_i$ where all $S_i$ are limited by $x \le S_i \le y $

Question

Sorry if this has already been asked or answered somewhere on the net.

I have a set of values $S=\{S_1,S_2,S_3,... S_n\} $ where $x \le S_i \le y $. S has an unknown number of discrete members, $S_i$, which can be set to any value we wish as long as it is within the range above. I understand intuitively that the minimum of $\sum_{i=1}^n=S_i$ should be $nx$, although I do not know how to go about proving it.

What is a simple way to prove that the minimum of the sum of S is $nx$?

Answer 1

So you want to minimize $\sum_{i=1}^nS_i=S_1+\cdots S_n$.

What is the minimum for $S_1$? $x$.

What is the minimum for $S_2$? $x$ again.

…

What is the minimum for $S_n$? still $x$.

So how many $x$ values do you have? $n$.

Since the minimum of a sum is the sum of the minimum values of its terms, it follows that the minimum is $nx$.

Answer 2

The condition $S_i \ge x$ for $i \in \{ 1, \dotsc, n \}$ implies $$ S = \sum_{i=1}^n S_i \ge \sum_{i=1}^n x = x \sum_{i=1}^n 1 = n x $$