Thread: use sequences to prove limit of |f(x)| HELP!

1. use sequences to prove limit of |f(x)| HELP!

Suppose f -> R and define the function |f|: D -> R by |f|(x) = |f(x)|. Prove that if lim as x -> xo f(x) = L then lim as x->xo |f|(x) exists and equals |L|.

I can do this using epsilons and deltas yet I am unsure as to how I would prove this using SEQUENCES.

2. Originally Posted by really.smarty
Suppose f -> R and define the function |f|: D -> R by |f|(x) = |f(x)|. Prove that if lim as x -> xo f(x) = L then lim as x->xo |f|(x) exists and equals |L|.

I can do this using epsilons and deltas yet I am unsure as to how I would prove this using SEQUENCES.
Let $\displaystyle x_n$ be any sequence converging to $\displaystyle x_0$. The fact that $\displaystyle \lim_{x\to x_0} f(x)= L$ tells you that $\displaystyle \lim_{n\to \infty} f(x_n)= L$. What can you say, then, about $\displaystyle \lim_{n\to \infty} |f(x_n)|$?

3. it would equal |L| but i am unsure as to how this will help me answer the question?

4. Hint, use the reverse triangle inequality, i.e. $\displaystyle | |x| - |y| | \leq |x-y|$