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. 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?

## more Combinatorics inquiries

Q1. The sequence$$\left$$ is
Q2. How numerous words have the right to be developed with the letter of words 'POSTMAN', if every word begins with T and also ends with M?
Q3. Find the amount of provided arithmetic progression 8 + 11 + 14 + 17 upto 15 state
Q4. The basic solution of recurrence relation$$a_r-5a_r-1+6a_r-2=4^r,\ r\ge2$$is:
Q5. The recurrence T(n) = 2T(n - 1) + n, because that n≥ 2 and T(1) = 1 evaluate to
Q6. If n pigeons space assigned come m pigeonholes then among the pigeonholes need to contain at the very least ______ pigeons.
Q7. If ai > 0 for ns = 1, 2, 3,..,n and also a1, a2, a3, ...an = 1 climate the biggest value the (1 + a1)(1 + a2)... (1 + an) is:
Q8. If m > 1 and also n∈ N, such the 1m+ 2m+ 3m+ ....+ nm>$$n \left( \dfracn+1k \right)^m$$, climate k = ?
Q9. Given the recurrence relationship f(n) = (n - 1) + f(n - 1), n > 72, f(2) = 1, then f(n) is:
Q10. Two makers are defective in a lot of 10. A mix of 4 machines is come be picked at a time native the lot. The maximum variety of combinations that can be acquired without any type of defective device is
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## more Engineering math Questions

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:
Q3. 60% of the employee of a company are college graduates. That these, 10% space in sales. Of the employees who did not graduate from college, 80% room in sales. The probability the an employee selected at random is in sales, is:
Q4. The probability density role of a random variable X is f(x) =$$\dfrac\pi10 sin \dfrac\pi x5$$; 0≤ x≤ 5. The very first quartile of X is:
Q5. If a discrete random variable X adheres to uniform distribution and also assume only the values 8, 9, 11, 15, 18, 20, the worth of P(|X - 14|
Q6. A Poisson circulation has a dual mode in ~ x = 1 and x = 2. The probability for x = 1 or for x = 2 of these 2 value is:
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 ?
Q9. Which one is parameter native population?
Q10. The excess Kurtosis the the Geometric circulation with parameter p is:
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