Every sequence in a closed and bounded collection S in succession Rn has actually a convergent subsequence (which converges come a suggest in S).

You are watching: Every bounded sequence has a convergent subsequence

Proof: Every succession in a closed and also bounded subset is bounded, so it has a convergent subsequence, which converges come a allude in the set because the set is closed.

Conversely, every bounded sequence is in a closed and bounded set, so it has a convergent subsequence.

Additional Information

AnotherBolzano-Weierstrasstheorem is:

Every bounded infinite set of genuine numbers contends least one limit point or cluster point.

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an ext Generating attributes Questions

Q1. The basic solution the recurrence relation\(a_r-5a_r-1+6a_r-2=4^r,\ r\ge2\)is:
Q2. The recurrence T(n) = 2T(n - 1) + n, for n≥ 2 and T(1) = 1 evaluates to
Q3. Given the recurrence relation f(n) = (n - 1) + f(n - 1), n > 72, f(2) = 1, climate f(n) is:
Q4. Every bounded succession has
Q5. Every bounded sequence has actually a cluster point; then this theorem is recognized as:
Q6. Solution to recurrence relationship T(n) = T(n - 1) + 2 is offered by, where n > 0 and also T(0) = 5.
Q7. The to run time of an algorithm is given byT(n) = T(n-1) + T(n-2) -T(n-3), if n > 3= n, otherwiseThen what need to be the relation between T(1), T(2) and T(3), so that the stimulate of the algorithm is continuous ?
Q8. The critical jobs are: A, D, H, and also I. What is the standard deviation that the task duration?Job A B C D E F G H IStandard deviation 1 4 0 1 0 1 2 1 1
Q9. The variety of permutations the the characters in LILAC so that no character appears in its initial position, if the 2 L's room indistinguishable, is ______.

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Q10. A girl has to make pizza with different toppings. There room 8 different toppings. In how numerous ways deserve to she make pizzas with 2 various toppings?

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Q1. The sequence\(\left\) is
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Q1. If the many correlation coefficient the X1 top top X2 and also X3 is zero, then:
Q2. If r12 = +0.80, r13 = -0.40 and r23 = -0.56, then the square of lot of correlation coefficient (correct to four decimal places)\(R^2_1._2_3\)is equal to:
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Q7. Which the the complying with is no a means of the sampling?
Q8. The 2nd and 4th moment around mean because that a circulation are 4 and 18 respectively. What is the worth of Pearson's coefficient that skewnessβz ?
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