Question

Please check my proof for error (involves limits of sequences):

Answer 1

Prove that if \[\lim |a _{n}| = 0 \implies \lim a_{n} =0\]

Let \[a_{n} \ge 0 = c_{n} , a_{n} \le 0 = b_{n}\]
then\[\lim |a_{n}| = \lim (a_{n}+b_{n}) \implies \lim c_{n} = \lim b_{n} \implies \lim a_{n} = 0\]

Answer 2

**\[ \lim (c_{n}+b_{n}) \]
in the last line **

Answer 3

Ok i need to re write the last line o.0 \[\lim |a_{n}| = \lim b_{n} + \lim c_{n} = \lim (b_{n}+c_{n}) = \lim a_{n}\]

Answer 4

I'm trying to understand whats going on. Are you splitting the sequence a_n into its positive and negative terms? Letting c_n be the sequence of all the positive, and b_n the sequence of all the negative?

Answer 5

yes

Answer 6

So for example, if the original sequence was:\[1,0,-2,0,3,0,-4,...\]thenthe sequence c_n would be:\[1,0,0,0,3,0,0\ldots\]and the sequence b_n would be:\[0,0,-2,0,0,0,-4,\ldots\]?

Answer 7

Yea, but its hard for me to see how the limit would be taken for those examples

Answer 8

ah, dont worry about the limit itself, I just wanted to make sure I understood your constuction of the new sequences c_n and b_n.

Answer 9

im convinced you do

Answer 10

I included the zeros thinking that they wouldn't effect the limits, i may have made a mistake in my understanding of the absolute value function. the question suggested the squeeze theorem be used... fyi

Answer 11

yeah, I can see using the squeeze theorem for sure. I'm trying to figure out how one can jump from:\[\lim_{n\rightarrow \infty}|a_n|=\lim_{n\rightarrow \infty}b_n+\lim_{n\rightarrow \infty}c_n\]I dont see the logic in that.

Answer 12

i think i may have misunderstood the absval function

Answer 13

it works like this:

|x|= x if x > 0 or x=0
|x| = -x if x<0

So the most you can do with the statement:\[|a_n|\]Is either note that:\[|a_n|=|b_n+c_n|\]or:\[|a_n|=c_n-b_n\]You have to make sure whatever on the right side remains positive. But i think the squeeze theorem would be a little more appropriate for this problem.

Answer 14

There is a property that the absolute value has. for any real number a, this inequality holds:\[-|a|\le a\le |a|\] If you put the sequence a_n in that statement, you get:\[-|a_n|\le a_n \le |a_n|\]

Answer 15

well theres the key!, i didnt know about that property.

Answer 16

thank you