Determine whether the following statements are true or not. Give a brief ex- planation. (e.g. proof,...

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Determine whether the following statements are true or not. Give a brief ex- planation. (e.g. proof, provide counter-examples etc.) In the following problems, let fn(r) be a sequence of functions on E 1. If fn(x) converges uniformly to f(x) on E, then f(x) converges pointwise to f(x) on E. 2. If f(x) converges pointwise to f(x) on E, then fn(r) converges uniformly on E. 3. Given a series of functions 1 fn(x), then its domain of convergence MUST be an interval. 4. The domain of convergence of power series is always an interval WITH BOTH ENDPOINTS. 5. Given a sequence of continuous functions fn(z) on E, then f(x) limn ofn (r) is also continuous on E. Determine whether the following statements are true or not. Give a brief ex- planation. (e.g. proof, provide counter-examples etc.) In the following problems, let fn(r) be a sequence of functions on E 1. If fn(x) converges uniformly to f(x) on E, then f(x) converges pointwise to f(x) on E. 2. If f(x) converges pointwise to f(x) on E, then fn(r) converges uniformly on E. 3. Given a series of functions 1 fn(x), then its domain of convergence MUST be an interval. 4. The domain of convergence of power series is always an interval WITH BOTH ENDPOINTS. 5. Given a sequence of continuous functions fn(z) on E, then f(x) limn ofn (r) is also continuous on E.

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