Prove that for all p E (0, co) there exist r, RE (0, co) such that...

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Prove that for all p E (0, co) there exist r, RE (0, co) such that the following set inclusions hold: ((x1, x2) E R²: x+x<²) C ((x1, x2) ER²: |x1|+|x2| ≤p} ((x1, x2) ER² |x1|+|x2| ≤p) C ((x1, x2) ER²x² + x ≤ R²). Hint: To prove the first inclusion, it might be useful to expand and rearrange the inequality (a - b)2 ≥0 for a, b = R. Prove that for all p E (0, co) there exist r, RE (0, co) such that the following set inclusions hold: ((x1, x2) E R²: x+x<²) C ((x1, x2) ER²: |x1|+|x2| ≤p} ((x1, x2) ER² |x1|+|x2| ≤p) C ((x1, x2) ER²x² + x ≤ R²). Hint: To prove the first inclusion, it might be useful to expand and rearrange the inequality (a - b)2 ≥0 for a, b = R.

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To prove the set inclusions we will first show that for any given p 0 we can find r 0 such that x x ... View the full answer