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Real Analysis: Show function defines a metric space and the space is complete

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Let X be the set of all continuous functions from I_1=[t_0-a_1, t_0+a_1] into the closed ball B[g(t_0);b] is a subset of R_n. Show that for each a>0 the rule
d(x,y)=max(|x(t)-y(t)|e^(-a|t-t_0|)) defines a metric on X and that the metric space (X,d) is complete.

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Completeness of a metric space is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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