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t00dles23
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Homework Statement
Let r be a positive number and define F = {u in R^n | ||u|| <= r}. Use the Componentwise Convergence Criterion to prove F is closed.
Homework Equations
The Componentwise Convergence Criterion states: If {uk} in F converges to c, then pi(uk) converges to pi(c). That is, the ith component of the sequence converges to the ith component of c.
The Attempt at a Solution
What we want to show is that if {uk} in F converges to c, then ||c|| <= r.
It's easy to show it without the componentwise convergence criterion, as follows:
||c|| <= ||c - uk|| + ||uk||. Taking the limit as k goes to infinity, we get ||c|| <= 0 + ||uk|| <= r. But I need a proof that does use the CCC.
I tried this:
Suppose {uk} in F converges to c. Then pi(uk) converges to pi(c) by the componentwise convergence criterion. So:
|pi(uk)| <= ||uk|| <= r
Taking the limit as k goes to infinity, we get
|pi(c)| <= r
(pi(c))^2 <= r^2
[tex]\sum_{i=1}^{n} (p_{i}(c))^2 \leq \sum_{i=1}^{n} r^2[/tex]
[tex]\sqrt{\sum_{i=1}^{n} (p_{i}(c))^2} \leq \sqrt{nr^2}[/tex]
||c|| <= (n)^(1/2)*r
but this isn't a strong enough statement, because I need ||c|| <= r. Advice please?