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SOLVED:Prove that in a Euclidean space, the operations of addition, multiplication by numbers and the formation of scalar products are all continuous. More exactly, prove that if x, →x, y, →y (in the sense of norm convergence) and A, →A (in the sense of ordinary convergence), then xn yn →x y, λn xn →λx, (xn, yn) →(x, y) Hint. Use Schwarz's inequality.

Problem 1 Show with an example that Euclid proof of the existence of a equilateral triangle does not work in the "rational plane Q? (that is denotes the subset of the Cartesian plane whose points have rational coordinates

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