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Suppose you have this two Lipschitz continuous functions:

$f_1$ ,with constant $L_1$ and $f_2$ with constant $L_2$.

I have to prove that $f_2 - f_1$ is Lipschitz continuous with constant $L_1+L_2$.

I did like this:

$|(f_2-f_1)(y)-(f_2-f_1)(x)|=|f_2(y)-f_2(x)-f_1(y)+f_1(x)|=$

$=|(f_2(y)-f_2(x))+(-f_1(y)+f_1(x))|$

By the triangle inequality:

$\leq|f_2(y)-f_2(x)|+|-f_1(y)+f_1(x)|$

$\leq L_2|y-x|+|-y+x|$

But I´m having problems in this last line,above...I should have $|y-x|$ in both,isn´t it?What´s wrong?how to proceed?Thanks?

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You are missing at $L_1$ on the last line of your formula. Also, $|z| = |-z|$, so $|-y+x| = |x-y| = |y-x|$. –  copper.hat Jul 12 '12 at 19:42
    
Thanks,too!!!!! –  Charlie Jul 12 '12 at 20:13

1 Answer 1

up vote 1 down vote accepted

One of the properties of a norm tells you that $|\lambda u|=|\lambda||u|$ for every scalar $\lambda$ and every vector $u$. Therefore $|-f_1(y)+f_1(x)|=|f_1(y)-f_1(x)|\le L_1|y-x|$ .

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Thanks!!!Very helpful!!! –  Charlie Jul 12 '12 at 19:39

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