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Handwritten and type work are both acceptable! I want it by the end of Nov 14th. Thank you!(1) Suppose f : E → R, p is a limit point of E ⊂ R, and lim f (x) > 0. Prove that there exist
x→p
c > 0 and δ > 0 such that f (x) > c when x ∈ E with 0 < |x − p| < δ. (2) Suppose f : E → R, p is a limit point of E, and lim f (x) exists. Prove that there exist x→p M > 0 and δ > 0 such that |f (x)| ≤ M when x ∈ E with 0 < |x − p| < δ. (3) Suppose f, g are real-valued functions on E ⊂ R and p is a limit point of E. If g is bounded on E and lim f (x) = 0, show that x→p lim f (x)g(x) = 0. x→p (4) Let f : (a, ∞) → R be such that lim xf (x) = L, L ∈ R. Prove that lim f (x) = 0. x→∞ x→∞ (5) Suppose f, g : E → R are continuous. Show that {x ∈ E : f (x) > g(x)} is open in E.
(6) Suppose f : [−1, 1] → R is continuous and satisfies f (−1) = f (1). Prove that there exists
γ ∈ [0, 1] such that f (γ) = f (γ − 1).
(7) Let f : E → R. Prove that f is continuous on E if and only if f −1 (F ) is closed in E for
every closed subset F ⊂ R.
(8) Let K ⊂ R be compact and f : K → R. Suppose that for each x ∈ K there exists εx > 0
such that f is bounded on K ∩ Nεx (x). Prove that f is bounded on K.
(9) Let f, g : E → R be uniformly continuous. If f and g are bounded, show that f g is
uniformly continuous on E.
(10) Suppose that E ⊂ R is bounded and f : E → R is uniformly continuous. Prove that f is
bounded on E.

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