Euclidean metric (L2 norm) versus taxicab metric(L1 norm)

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The discussion focuses on proving that the Euclidean metric (L2 norm) is always less than or equal to the taxicab metric (L1 norm) for any vector x in R^n. The individual seeks a method to demonstrate this relationship, noting that it appears obvious but requires formal proof. The proposed approach involves showing that the expression (||x||1)² - (||x||2)² is never negative. This suggests a mathematical exploration of the properties of these norms to establish the inequality. The conversation centers on the mathematical rigor needed to validate this comparison between the two metrics.
mglaros
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Homework Statement



I was just wondering how I would go about proving that the euclidean metric is always smaller than or equal to the taxicab metric for a given vector x in R^n. The result seems obvious but I am not sure how I would show this.

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The Attempt at a Solution

 
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Show that (||x||1)2 - (||x||2)2 can never be negative.
 

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