Let a, b ∈ R with a < b and let f : [a, b] -> R be continuous. State the extreme value theorem for f. You may assume that y = sup [a,b] f is a finite real number.
Justify why, for any e > 0, there exists 1 ∈ [a, b] such that y - e ≤ f(x) ≤ y.
Hence show that there exists a sequence (Xn)n=1 ^[infinity] in (a, b) such that f(xn) → y as n → [infinity] Justify the existence of a convergent sequence (2n)n=1 ^[infinity] in [a,b] such that f(2n) → y
Hence conclude that there exists r ∈ (a, b) such that f(x) = y.