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Bounded Real Sequences and Metrics

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Let X be the set of all bounded real sequences ... (SEE ATTACHMENT FOR COMPLETE QUESTION) ... For points x = {x1,,x2,x3...} and y = {y1,y2,y3...} in X define:
d(x,y) := SUP |xk-yk|.
(k less than or equal to 1)
Prove that d is a metric (remember to explain why d(x,y) is finite!)

(Please explain in your own words how the proof works. If you use a theorem, please state what it is and if possible, where you got it).

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Bounded Real Sequences and Metrics are investigated.

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First of all, we will show that d(x,y) = finite.
Let Mx >0 so that

Now, we need to check all the axioms of the metric:
This is obvious because of the properties of module:

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