[ATTACH=full]3282[/ATTACH]
63. https://www.math-forums.com/threads/real-numbers-a-b-c.441909/#post-1099802 somewhere must be 62 too, or use 63 to solve 62 in similar way
62. show that for all real numbers a and b, we have |a| - |b| <= |a-b| we start with what we know: |a|+|b| ≥|a−b| Now, substitute a=a and b=a−b to get, |a−b|≥|b|−|a| ......................(1) Also, we can writ |a|+|b|≥|b−a| as |a−b|=|−(a−b)|=|b−a| Now, substitute a=b−a and b=b to get, |b−a| ≥ |a|−|b|....................(2) Now, as |b−a|=|a−b| We may write (2) as |a−b|≥|a|−|b|.............(3) because |a−b|=|−(a−b)|=|b−a| From (1) and (3), we can conclude that |a−b| ≥ ||a|−|b|| or vice verse |a|−|b|| ≤|a−b|
Very cool. I will study this proof when time allows. My weekend is over just like that. I graduated from high school in 1984. I then took a few steps after graduation and find myself at 57 years old. How on earth did time go by so fast?