Answer to Question #123528 in Real Analysis for Tau

Question #123528
Show,The set U = f(x; y) 2 R2 j x2 + y2 <=1, and x > 0g is open in B(0; 1)
where the norm on R2 is the Euclidean norm.
1
Expert's answer
2020-06-23T20:16:19-0400

As, the question is not clearly defined, I am assuming the question goes like this otherwise it doesn't make sense.

"U=\\{(x,y)\\in \\mathbb{R}^2:x^2+y^2\\leq 1\\&x>0\\}"

We have to show "U" is open in "\\overline{B}(0,1)" where "B(0,1)" is open ball centered at with radius "1" and of course norm is Euclidean norm.

Clearly, our induced matrix space is "\\overline{B}(0,1)" where metric is induced from "(\\mathbb{R}^2,|| \\: ||_2)"

Let, for any

"v_n=(x_n,y_n)\\in U"

consider "r_n=1-\\frac{1}{n}>0" ,Thus,consider the open ball "B'(v_n,r_n)" ,Hence

"U=\\cup_{n=1}^{\\infty}B'(v_n,r_n)"

Thus, we are done.


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Comments

Assignment Expert
24.06.20, 22:38

Dear Tau, You are welcome. We are glad to be helpful. If you liked our service, please press a like-button beside the answer field. Thank you!

Tau
24.06.20, 15:45

You have amazing expert who even can understand the mind what the question would be! Thank you very much assignmentexpert for your help.

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