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Does triangle inequality hold for summations and sup?

1,100
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I know that the triangle inequality is lx+y|<= |x| +|y|

Does this hold for a summation of say Sigma (|xn+yn|) <= sigma|xn| + sigma|yn| for n=0 to infinity?

would this also work for sup|x+y| ??
 

Answers and Replies

LeonhardEuler
Gold Member
858
1
gravenewworld said:
I know that the triangle inequality is lx+y|<= |x| +|y|

Does this hold for a summation of say Sigma (|xn+yn|) <= sigma|xn| + sigma|yn| for n=0 to infinity?
Yes:
[tex]\sum_{n=0}^{N}{|x_n+y_n|}[/tex]
[tex]\leq |x_0| + |y_0| + \sum_{n=1}^{N}{|x_{n}+y_n|}[/tex]
[tex]\leq |x_0| + |y_0| + |x_1| + |y_1| + \sum_{n=2}^{N}{|x_{n}+y_n|}[/tex]
...
[tex]\leq \sum_{n=0}^{N}{|x_n|} +\sum_{n=0}^{N}{|y_n|}[/tex]
 
Last edited:
AKG
Science Advisor
Homework Helper
2,559
3
LeonhardEuler, the original poster asked about infinite sums, not sums to N (which I assume you used to represent some natural number). If the sum is infinite, then if xn = 1 for all n, and yn = -1 for all n, then [itex]\sum _{n=0} ^{\infty} |x_n|[/itex] and [itex]\sum _{n=0} ^{\infty} |y_n|[/itex] aren't even defined.

Suppose all the series in question do in fact converge. Then suppose the desired inequality is not true, then we'd have:

[tex]\sum _{n = 0} ^{\infty} |x_n + y_n| > \sum _{n=0} ^{\infty} |x_n| + \sum _{n=0} ^{\infty} |y_n|[/tex]

[tex]\lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n + y_n|\right ) > \lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n|\right ) + \lim _{N \to \infty} \left ( \sum _{n = 0} ^N |y_n|\right )[/tex]

[tex]\lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n + y_n|\right ) > \lim _{N \to \infty} \left ( \sum _{n = 0} ^N |x_n| + \sum _{n = 0} ^N |y_n|\right )[/tex]

thus there is some N such that:

[tex]\sum _{n = 0} ^N |x_n + y_n| > \sum _{n = 0} ^N |x_n|\right + \sum _{n = 0} ^N |y_n|[/tex]

which LeonhardEuler has proven false.
 
1,100
25
alright thanks guys!
 

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