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Absolute Values and Inequality understanding

  1. Jun 21, 2010 #1
    1. The problem statement, all variables and given/known data

    I saw this in my real analysis text book and I have been trying to understand how this equation [tex]\left | x-c \right |< 1 [/tex]

    you can get this:[tex] \left | x \right |\leq \left | c \right | + 1[/tex]

    2. Relevant equations

    I wanted to know what steps made this possible , particularly why it changed from [tex]< to \leq [/tex]

    3. The attempt at a solution
    My thinking was that, they did this because it makes no difference as to if you put [tex]< or \leq [/tex]

    Any help is appreciated!
    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 21, 2010 #2
    [tex]|x|-|c|\leq |x-c| < 1 \Rightarrow |x|-|c| \leq 1[/tex]
    This one is true because of the symbol [tex]\Rightarrow [/tex] (or as you said, it makes no difference). Things would be different if it were [tex]\Leftrightarrow[/tex].
     
  4. Jun 21, 2010 #3
    You forgot to tell the young man that to understand this problem he also needs to understand two of the three basic norms Homogeneity and Subadditivity
     
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