Let f(adsbygoogle = window.adsbygoogle || []).push({}); _{n}: R [tex]\rightarrow[/tex] R be the function

f_{n}= [tex]\frac{1}{n^3 [x-(1/n)]^2+1}[/tex]

Let f : R [tex]\rightarrow[/tex] R be the zero function.

a. Show that f_{n}(x) [tex]\rightarrow[/tex] f(x) for each x [tex]\in[/tex] R

b. Show that f_{n}does not converge uniformly to f. (This shows that the converse of Theorem 21.6 does not hold; the limit function f may be continuous even though the convergence is not uniform.)

a. i'm not sure...

is f(x) equivalent to f_{1}(x)?

If it is.... then... the function would be...

[tex]\frac{1}{[x-1]^2+1}[/tex]

[tex]\frac{1}{x^2-2x+2}[/tex]

but i'm not sure how to show f_{n}(x) [tex]\rightarrow[/tex] f(x) for each x [tex]\in[/tex] R

b. Theorem 21.6 states,

" let f_{n}: X[tex]\rightarrow[/tex] Y be a sequence of continuous functions from the topological space X to the metric space Y. If (f_{n}) converges uniformly to f, then f is continuous. "

The converse of this is...

"If f is continuous, then (f_{n}) converges uniformly to f. "

i don't know how to prove a function is not convergent.

Can someone help me?

Thank You

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# Convergence on R->R

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