Why Does The Squeeze Theorem Justify Limits Of Upper And Lower Sums?

[Bashyboy]

I am currently reading about finding areas under graphs using summations, specifically taking the of the number of rectangles, n, goes to infinity. My books says that "because the same limit value is attained for both minimum value f(m_i) and the maximum value f(M_i), it follows from the squeeze theorem that the choice of x in the ith sub-interval does not affect the limit." Why does the squeeze theorem say that I can do this?

 

[Bashyboy]

I suppose another way of wording my question is why can I choose any x-value in the sub-interval.

 

[algebrat]

Because f(m_i)≤f(x_i)≤f(M_i).

Then summing, the middle sum is squeezed by the other two sums.