Absolute Values and Inequality understanding

[jzapata87]

Homework Statement

I saw this in my real analysis textbook and I have been trying to understand how this equation \left | x-c \right |< 1

you can get this:\left | x \right |\leq \left | c \right | + 1

Homework Equations

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

The Attempt at a Solution

My thinking was that, they did this because it makes no difference as to if you put < or \leq

Any help is appreciated!
Thanks

 

[hikaru1221]

|x|-|c|\leq |x-c| < 1 \Rightarrow |x|-|c| \leq 1
This one is true because of the symbol \Rightarrow (or as you said, it makes no difference). Things would be different if it were \Leftrightarrow.

 

[Susanne217]

|x|-|c|\leq |x-c| < 1 \Rightarrow |x|-|c| \leq 1
This one is true because of the symbol \Rightarrow (or as you said, it makes no difference). Things would be different if it were \Leftrightarrow.

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