1. ## Inequality involving modulus

This is a simple step in a proof but I can't see why it works (sorry, I'm no good at dealing with modulus sign...). Here's the step, if anyone could explain it to me that would be great!

$\displaystyle |\frac{a}{b}-c | \leq \epsilon \Rightarrow | \frac{a}{b} | \leq \epsilon + |c|$

Thanks

2. ## Re: Inequality involving modulus

Originally Posted by Ant
This is a simple step in a proof but I can't see why it works (sorry, I'm no good at dealing with modulus sign...). Here's the step, if anyone could explain it to me that would be great!

$\displaystyle |\frac{a}{b}-c | \leq \epsilon \Rightarrow | \frac{a}{b} | \leq \epsilon + |c|$
Just take note of the fact $\displaystyle |x|-|y|\le |x-y|$ for all $\displaystyle x~\&~y$ in $\displaystyle \mathbb{R}$.

3. ## Re: Inequality involving modulus

Originally Posted by Plato
Just take note of the fact $\displaystyle |x|-|y|\le |x-y|$ for all $\displaystyle x~\&~y$ in $\displaystyle \mathbb{R}$.
So we have:

$\displaystyle |\frac{a}{b}| - |c| \leq |\frac{a}{b}-c| \leq \epsilon \Rightarrow |\frac{a}{b}| - |c| \leq \epsilon \Rightarrow | \frac{a}{b} | \leq \epsilon + |c|$

4. ## Re: Inequality involving modulus

Originally Posted by Ant
So we have:

$\displaystyle |\frac{a}{b}| - |c| \leq |\frac{a}{b}-c| \leq \epsilon \Rightarrow |\frac{a}{b}| - |c| \leq \epsilon \Rightarrow | \frac{a}{b} | \leq \epsilon + |c|$
You got it! Way to go.