i HAVE DIFFICULTIES WIH AN ASSIGMENT. ANY HELP ?

2A

Let M be a set of non-negative real valued function on the real valued axe with

M = { f: R→ R | f(x) ≥ 0 for alle x belongs to R }

Show that the matric distance

d(f,g) = supp {|e^(-f(x)) -e ^-g(x)|, | x belongs to R}, f,g belongs to R

Defines a metric d on the set M

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2B

Let for n belongs to N, the functions fn and gn belongs to M, n belongs to N be defined by

fn(x) = x^2 if |x| ≤ n

n^2 if |x| > n

and

gn(x) = x^2 if |x| ≤ n

= 0 if |x| > n

Do the functional series fn(x) (n belongs to N)

and the functional series gn(x) (n belongs to N) converges uniformly on R ?

If not unifrom convergent for what reasons are the functions series fn and gn not unifrom convergent ?

For each of the two functions determine its convergence on the metric space (M,d) ?

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BR

mathcph

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