Does triangle inequality hold for summations and sup?

[gravenewworld]

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| ??

 

[LeonhardEuler]

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:
$$\sum_{n=0}^{N}{|x_n+y_n|}$$
$$\leq |x_0| + |y_0| + \sum_{n=1}^{N}{|x_{n}+y_n|}$$
$$\leq |x_0| + |y_0| + |x_1| + |y_1| + \sum_{n=2}^{N}{|x_{n}+y_n|}$$
...
$$\leq \sum_{n=0}^{N}{|x_n|} +\sum_{n=0}^{N}{|y_n|}$$

 

[AKG]

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 x_n = 1 for all n, and y_n = -1 for all n, then $\sum _{n=0} ^{\infty} |x_n|$ and $\sum _{n=0} ^{\infty} |y_n|$ 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:

$$\sum _{n = 0} ^{\infty} |x_n + y_n| > \sum _{n=0} ^{\infty} |x_n| + \sum _{n=0} ^{\infty} |y_n|$$

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

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

thus there is some N such that:

$$\sum _{n = 0} ^N |x_n + y_n| > \sum _{n = 0} ^N |x_n|\right + \sum _{n = 0} ^N |y_n|$$

which LeonhardEuler has proven false.

 

[gravenewworld]

alright thanks guys!