Lebl-Basic-Analysis-I · Problem 7.3.14

Metric Spaces

Mathematics · sequences-and-convergence

Problem

Consider Rn\mathbb{R}^n, and let dd be the standard euclidean metric. Let d′(x,y)≔∑ℓ=1n∣xℓ−yℓ∣d'(x, y) \colonequals \sum_{\ell=1}^n |x_\ell - y_\ell| and d′′(x,y)≔max⁡{∣x1−y1∣,∣x2−y2∣,⋯ ,∣xn−yn∣}d''(x, y) \colonequals \max\{|x_1 - y_1|, |x_2 - y_2|, \cdots, |x_n - y_n|\}. a) Use Exercise 7.1.6, to show that (Rn,d′)(\mathbb{R}^n, d') and (Rn,d′′)(\mathbb{R}^n, d'') are metric spaces. b) Let {xk}k=1∞\{x_k\}_{k=1}^\infty be a sequence in Rn\mathbb{R}^n and p∈Rnp \in \mathbb{R}^n. Prove that the following statements are equivalent: ((1)) {xk}k=1∞\{x_k\}_{k=1}^\infty converges to pp in (Rn,d)(\mathbb{R}^n, d). ((2)) {xk}k=1∞\{x_k\}_{k=1}^\infty converges to pp in (Rn,d′)(\mathbb{R}^n, d'). ((3)) {xk}k=1∞\{x_k\}_{k=1}^\infty converges to pp in (Rn,d′′)(\mathbb{R}^n, d'').

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