Where does that in the center come from? Darbou amounts?

Question

Mathematics

How do we arrive at what is highlighted. Please explain. Something stupid

Answer 1

S − s < ε ⇒ S < s + ε
s ≤ I ≤ S ⇒ s ≤ I < s + ε
s ≤ σ ≤ S ⇒ s ≤ σ < s + ε
|σ − I| = σ − I, for σ ≥ I
|σ − I| = I − σ, for σ ≤ I
max(σ − I) = max(σ) − min(I) = (s + ε) − s = ε
max(I − σ) = max(I) − min(σ ) = (s + ε) − s = ε
max(|σ − I|) = max(max(σ − I), max(I − σ)) = ε
|σ − I| < ε

Answer 2

Answer 3

So it is written there,
s <= I <= S
s <= sigma <= S
turns out both I and sigma, are in the range between s and S and cannot be less than s, nor more than S
and times S - s < epsilon, then the difference sigma - I does not exceed epsilon modulo