Sequence of functions

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April 3rd 2011, 04:06 PM
Connected

Sequence of functions

Consider the sequence of functions $f_n(t)=\cos^{2n}(n!\pi t),$ on the space $(\mathcal C[0,1],d_\infty).$ Is $f_n(t)$ convergent?

Well I think this is a limit problem, but I don't know how to compute it, any ideas?
April 3rd 2011, 04:53 PM
Drexel28

Quote:

Originally Posted by Connected

Consider the sequence of functions $f_n(t)=\cos^{2n}(n!\pi t),$ on the space $(\mathcal C[0,1],d_\infty).$ Is $f_n(t)$ convergent?

Well I think this is a limit problem, but I don't know how to compute it, any ideas?

No. Note that if it was convergent then $\displaystyle \lim f_n(t)$ would be continuous...yet I don't think it's hard to show that $\lim f_n(t)=\mathbf{1}_{\mathbb{Q}}$
April 3rd 2011, 05:21 PM
Connected

Quote:

Originally Posted by Drexel28

yet I don't think it's hard to show that $\lim f_n(t)=\mathbf{1}_{\mathbb{Q}}$

What do you mean with the right notation?

How do you solve the problem? :D
April 3rd 2011, 05:23 PM
Drexel28

Quote:

Originally Posted by Connected

What do you mean with the right notation?

How do you solve the problem? :D

It is the indicator function--and no I can't show you how to solve the problem. This is your homework, me telling you how to solve it will mean that you might fail your test, drop out of school, etc. I can't handle that kind of guilt.

Why don't you solve it, and we'll check it.
April 3rd 2011, 05:27 PM
Connected

But I'm not actually asking you to solve me the problem completely, a hint could be a help. :(
April 3rd 2011, 05:37 PM
Drexel28

Quote:

Originally Posted by Connected

But I'm not actually asking you to solve me the problem completely, a hint could be a help. :(

I gave you a big hint. If $f_n\to f$ in $C[0,1]$ then $f$ must be continuous, but a quick check shows that $f_n\to\mathbf{1}_{\mathbb{Q}}$ which is the indicator function on the rationals.

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