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{Sn} is convergent -> {|Sn|} is convergent

  • Thread starter Jamin2112
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  • #1
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{Sn} is convergent ---> {|Sn|} is convergent

Homework Statement



I need to prove that if {sn} is convergent, then {|sn|} is convergent.

Homework Equations



sn is convergent if for some s and all ε > 0 there exists a positive integer N such that |sn - s| < ε whenever nN.

The Attempt at a Solution



Proof. By contrapositive. Suppose {|sn|} is not convergent. Then for all s there exists an ε > 0 such that ||sn| - s|| ≥ ε for all n.

..... I need to somehow show that this implies that {sn} does not converge.

Maybe some fancy triangle inequality thing like

ε ≤ ||sn| - s| ≤ ||sn| - sn| + |sn - s|

Wat do, PF?
 
Last edited:

Answers and Replies

  • #2
986
9


Ah, wait. I think I figured out something.

ε ≤ ||sn| - s| ≤ ||sn| - sn| + |sn - s| ≤ 2|sn| + |sn - s|

Am I any closer?
 
Last edited:
  • #3
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You are supposed to show that |Sn| converges to |S| not to S.
The trick is to show that ||Sn|-|S|| <= |Sn-S|

Hint to showing it:
|S| = |Sn + (S-Sn)|
And use the triangle inequality.
 
  • #4
986
9


You are supposed to show that |Sn| converges to |S| not to S.
The trick is to show that ||Sn|-|S|| <= |Sn-S|

Hint to showing it:
|S| = |Sn + (S-Sn)|
And use the triangle inequality.
I knew the triangle inequality would show his face somewhere.
 
  • #5
986
9


screen-capture-57.png




http://images.icanhascheezburger.com/completestore/2009/3/5/128807888821558038.jpg
 
  • #6
986
9


Hey, I need a hint for another problem. (You guys give stellar hints. Pat yourselves on the back.)

Suppose {pn} is a Cauchy sequence in a metric space X, and some subsequence {pni} converges to a point p in X. Prove that the full sequence {pn} converges to p.

As always, I'd love to use contradiction.

Suppose {pn} does not converge to p. Then for some ε > 0, d(p,pn) ≥ ε for all n. Then d(pm,pn)+d(pm,p) ≥ ε for all m, n. But {pm} is a Cauchy sequence, so with the proper choice of m, n we'll have d(pm,pn) < ε. ........ blah blah I hope this is going somewhere. A little hint, maybe?
 

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