Statement 1 | Suppose ∑|a_i| diverges and ∑ a_i = 2. There is a rearrangement a_i_k of the terms such that ∑ a_i_k = 4. Statement 2 | There exists metric spaces X and Y with X closed and bounded and a continuous mapping f : X → Y such that f(X) is NOT “closed and bounded”.

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∻Solution by Steps∻ ‖ step 1 ⋮ Consider the series ∑|a_i| and ∑a_i. If ∑|a_i| diverges, it means that the series of absolute values does not have a finite sum. ‖ ‖ step 2 ⋮ Given that ∑a_i = 2, it implies that the series of a_i converges to 2. ‖ ‖ step 3 ⋮ By the Riemann series theorem, if a series is conditionally convergent (which means ∑a_i converges but ∑|a_i| diverges), then its terms can be rearranged to converge to any real number. ‖ ‖ step 4 ⋮ Therefore, there exists a rearrangement ∑a_i_k of the series such that ∑a_i_k = 4. ‖ ∻Answer∻ ⚹ Statement 1 is true. ⚹ ∻Key Concept∻ ⚹Riemann Series Theorem⚹ ∻Explanation∻ ⚹The Riemann series theorem states that a conditionally convergent series can be rearranged to sum to any real number.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ Consider metric spaces X and Y, and a continuous mapping f: X → Y. ‖ ‖ step 2 ⋮ If X is closed and bounded, it does not necessarily imply that f(X) is closed and bounded in Y. ‖ ‖ step 3 ⋮ A counterexample can be constructed using the space X = [0, 1] and the space Y = ℝ (the set of all real numbers) with a continuous function that maps X onto an unbounded set in Y. ‖ ‖ step 4 ⋮ For instance, the function f(x) = tan(πx/2) maps the closed and bounded interval [0, 1) onto the entire set of real numbers, which is not bounded. ‖ ‖ step 5 ⋮ Therefore, there exists a continuous mapping from a closed and bounded space to another space that is not closed and bounded. ‖ ∻Answer∻ ⚹ Statement 2 is true. ⚹ ∻Key Concept∻ ⚹Continuous Functions and Metric Spaces⚹ ∻Explanation∻ ⚹The image of a closed and bounded set under a continuous function is not necessarily closed and bounded.⚹◊Can you provide an example of a rearrangement of terms in a series where the sum of the terms changes from 2 to 4?⍭ Generate me a similar question◊

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