Thread: Converging to an absolute Value

1. Converging to an absolute Value

Prove that if {a} converges to A, then {abs value(a)} converges to absolute value(A). I want to know if the converse si true(What does that even mean?).

Here is what I have:
epsilon>0 is arbitrary
A-epsilon<a<A+epsilon

Don't know where to go with the absolute value. i'm confused....

2. This is really simple!
$\left| {\left| {a_n } \right| - \left| A \right|} \right| \leqslant \left| {a_n - A} \right| < \varepsilon$.

Consider $b_n = (-1)^n$ then $\left| {b_n } \right| \to 1$

3. Originally Posted by Plato
This is really simple!
$\left| {\left| {a_n } \right| - \left| A \right|} \right| \leqslant \left| {a_n - A} \right| < \varepsilon$.
Ok, so then I would take off the absolute values and find n>N, right?

4. Also, what does it mean when it asks if the converse is true?

5. Originally Posted by kathrynmath
Also, what does it mean when it asks if the converse is true?
Does that mean if {absolute value(a)} converges to absolute value(A) then {a} converges to A?

6. Originally Posted by kathrynmath
Does that mean if {absolute value(a)} converges to absolute value(A) then {a} converges to A?
yes

7. Originally Posted by kathrynmath
Does that mean if {absolute value(a)} converges to absolute value(A) then {a} converges to A?
No it does not mean that.
Consider: $b_n = \left( { - 1} \right)^n \Rightarrow \quad \left( {\left| {b_n } \right|} \right) \to 1\,\& \,\left( {b_n } \right) \to ?$

8. Originally Posted by kathrynmath
Does that mean if {absolute value(a)} converges to absolute value(A) then {a} converges to A?
Originally Posted by Jhevon
yes
Come on you know better than that. It is not true.

9. Originally Posted by Plato
Come on you know better than that. It is not true.
i thought post #5 refered to post #4 which was talking about the problem in post #1

in which case the answer is yes. asking if the converse is true is the same as asking what post #5 said

10. Originally Posted by Jhevon
i thought post #5 refered to post #4 which was talking about the problem in post #1

in which case the answer is yes. asking if the converse is true is the same as asking what post #5 said
yeah, that's what I meant.

11. Originally Posted by kathrynmath
yeah, that's what I meant.
yes, that's what i assumed you meant

i was answering your question as to what you need to prove (or in this case, disprove) not answering the actual problem, Plato did that, and gave a nice counter example

12. Originally Posted by Jhevon
i thought post #5 refered to post #4 which was talking about the problem in post #1
in which case the answer is yes. asking if the converse is true is the same as asking what post #5 said