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.

Purchase answer to see full

attachment

#### Why Choose Us

- 100% non-plagiarized Papers
- 24/7 /365 Service Available
- Affordable Prices
- Any Paper, Urgency, and Subject
- Will complete your papers in 6 hours
- On-time Delivery
- Money-back and Privacy guarantees
- Unlimited Amendments upon request
- Satisfaction guarantee

#### How it Works

- Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
- Fill in your paper’s requirements in the "
**PAPER DETAILS**" section. - Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
- Click “
**CREATE ACCOUNT & SIGN IN**” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page. - From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.