Lesson test: Limits, proved

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Limits, proved

1.[2p]
Put the lines of the proof that $\frac{2 n + 3}{n + 1} \to 2$ in order.

Since $ε$ was arbitrary, $a_{n} \to 2$ .
Let $ε > 0$ .
Then $| a_{n} - 2 | = \frac{1}{n + 1} < \frac{1}{n} \leq \frac{1}{N} < ε$ .
By the Archimedean property there is $N \in ℕ$ with $N > \frac{1}{ε}$ .
Let $n \geq N$ .

Show the answer

a, b, c, d, e

2.[2p]
For $a_{n} = \frac{3 n + 1}{n + 2}$ and $L = 3$ , what is the smallest $N$ such that $| a_{n} - 3 | < 0.01$ for every $n \geq N$ ?

CorrectNot quite: 499

3.[1p]
For $a_{n} = \frac{1}{n}$ and $L = 0$ , what is the smallest $N$ such that $| a_{n} | < 0.003$ for every $n \geq N$ ?

CorrectNot quite: 334

4.[1p]
In the proof that $\frac{1}{n} \to 0$ , which step needs the Archimedean property?

Finding a natural number $N$ with $N > \frac{1}{ε}$

Correct

Showing that $\frac{1}{n} \leq \frac{1}{N}$ when $n \geq N$

The answer is: Finding a natural number $N$ with $N > \frac{1}{ε}$

Showing that $\frac{1}{n}$ is positive

The answer is: Finding a natural number $N$ with $N > \frac{1}{ε}$

Letting $ε$ be an arbitrary positive number

The answer is: Finding a natural number $N$ with $N > \frac{1}{ε}$

5.[2p]
A proof that $\frac{3 n + 1}{n + 2} \to 3$ says "take $N = \frac{5}{ε}$ ". What would a careful reader object to?

$\frac{5}{ε}$ is usually not a natural number, so it cannot serve as $N$

Correct

$N$ is not allowed to depend on $ε$

The answer is: $\frac{5}{ε}$ is usually not a natural number, so it cannot serve as $N$

$N$ must be the smallest position that works

The answer is: $\frac{5}{ε}$ is usually not a natural number, so it cannot serve as $N$

The proof should have started from $| a_{n} - 3 | < ε$ and worked backwards

The answer is: $\frac{5}{ε}$ is usually not a natural number, so it cannot serve as $N$

6.[2p]
Which of these are correct about proving that $(- 1)^{n}$ diverges?

Select all that apply

The proof must show the sequence converges to no real number $L$ , so it starts with an arbitrary $L$

Correct

The prover chooses the tolerance, here $ε = 1$

Correct

For each $N$ , one of the terms at positions $2 N$ and $2 N + 1$ lies at least $1$ from $L$

Correct

It is enough to show that the sequence does not converge to $1$

The answer is: The proof must show the sequence converges to no real number $L$ , so it starts with an arbitrary $L$ , The prover chooses the tolerance, here $ε = 1$ , For each $N$ , one of the terms at positions $2 N$ and $2 N + 1$ lies at least $1$ from $L$

The sequence diverges because it is unbounded

7.[3p]
Match each theorem to the move its proof turns on.

Uniqueness of limits
Limit of a sum
Convergent sequences are bounded
Monotone convergence

the tolerance $\frac{| L - M |}{2}$ and a contradiction
the supremum of the terms, from completeness
tolerance $1$ and the finitely many earlier terms
tolerance $\frac{ε}{2}$ for each sequence

Show the answer

Uniqueness of limits: the tolerance $\frac{| L - M |}{2}$ and a contradiction Limit of a sum: tolerance $\frac{ε}{2}$ for each sequence Convergent sequences are bounded: tolerance $1$ and the finitely many earlier terms Monotone convergence: the supremum of the terms, from completeness

8.[1p]
Every bounded sequence of real numbers converges.

True

The answer is: False

False

Correct

9.[3p]
Which of these are true of the decimal truncations $1, 1.4, 1.41, 1.414, \dots$ of $\sqrt{2}$ ?

Select all that apply

Every term is rational and the sequence is increasing and bounded above

Correct

In $ℝ$ the sequence converges to $\sqrt{2}$

Correct

In $ℚ$ the sequence has no limit

Correct

In $ℚ$ the proof of monotone convergence breaks where it takes a supremum

Correct

The sequence diverges in $ℝ$ because $\sqrt{2}$ is irrational

The answer is: Every term is rational and the sequence is increasing and bounded above, In $ℝ$ the sequence converges to $\sqrt{2}$ , In $ℚ$ the sequence has no limit, In $ℚ$ the proof of monotone convergence breaks where it takes a supremum

10.[3p]
Which of these conclusions follow from the algebra of limits, given only that $a_{n} \to 2$ and $b_{n} \to 0$ ?

Select all that apply

$a_{n} + b_{n} \to 2$

Correct

$a_{n} b_{n} \to 0$

Correct

$\frac{b_{n}}{a_{n}} \to 0$

Correct

$\frac{a_{n}}{b_{n}} \to \infty$

The answer is: $a_{n} + b_{n} \to 2$ , $a_{n} b_{n} \to 0$ , $\frac{b_{n}}{a_{n}} \to 0$ , $3 a_{n} - b_{n} \to 6$

$3 a_{n} - b_{n} \to 6$

Correct

11.[2p]
To prove $\frac{(- 1)^{n}}{n} \to 0$ by the squeeze theorem, which pair of bounding sequences works?

$- \frac{1}{n} \leq \frac{(- 1)^{n}}{n} \leq \frac{1}{n}$

Correct

$- 1 \leq \frac{(- 1)^{n}}{n} \leq 1$

The answer is: $- \frac{1}{n} \leq \frac{(- 1)^{n}}{n} \leq \frac{1}{n}$

$0 \leq \frac{(- 1)^{n}}{n} \leq \frac{1}{n}$

The answer is: $- \frac{1}{n} \leq \frac{(- 1)^{n}}{n} \leq \frac{1}{n}$

$\frac{(- 1)^{n}}{n} \leq (- 1)^{n}$

The answer is: $- \frac{1}{n} \leq \frac{(- 1)^{n}}{n} \leq \frac{1}{n}$

12.[1p]
Every bounded sequence of real numbers has a convergent subsequence.

True

Correct

False

The answer is: True

13.[2p]
In proving that $f (x) = 3 x - 1$ is continuous at $a = 2$ , what is the largest $δ$ that works for $ε = 0.3$ ?

CorrectNot quite: 0.1

14.[2p]
Why is $H (x) = 0$ for $x < 0$ , $H (x) = 1$ for $x \geq 0$ not continuous at $0$ ?

With $ε = \frac{1}{2}$ , every $δ > 0$ is defeated by $x = - \frac{δ}{2}$

Correct

$H$ is not defined at $0$

The answer is: With $ε = \frac{1}{2}$ , every $δ > 0$ is defeated by $x = - \frac{δ}{2}$

$lim_{x \to 0} H (x) = 1$ but $H (0) = 0$

The answer is: With $ε = \frac{1}{2}$ , every $δ > 0$ is defeated by $x = - \frac{δ}{2}$

Every $δ > 0$ is defeated when $ε = 2$

The answer is: With $ε = \frac{1}{2}$ , every $δ > 0$ is defeated by $x = - \frac{δ}{2}$

1.[2p] Put the lines of the proof that 2n+3n+1→2 in order.

2.[2p] For an=3n+1n+2 and L=3, what is the smallest N such that |an−3|<0.01 for every n≥N?

3.[1p] For an=1n and L=0, what is the smallest N such that |an|<0.003 for every n≥N?

4.[1p] In the proof that 1n→0, which step needs the Archimedean property?

5.[2p] A proof that 3n+1n+2→3 says "take N=5ε". What would a careful reader object to?

6.[2p] Which of these are correct about proving that (−1)n diverges?

7.[3p] Match each theorem to the move its proof turns on.

8.[1p] Every bounded sequence of real numbers converges.

9.[3p] Which of these are true of the decimal truncations 1,1.4,1.41,1.414,… of 2?

10.[3p] Which of these conclusions follow from the algebra of limits, given only that an→2 and bn→0?

11.[2p] To prove (−1)nn→0 by the squeeze theorem, which pair of bounding sequences works?

12.[1p] Every bounded sequence of real numbers has a convergent subsequence.

13.[2p] In proving that f(x)=3x−1 is continuous at a=2, what is the largest δ that works for ε=0.3?

14.[2p] Why is H(x)=0 for x<0, H(x)=1 for x≥0 not continuous at 0?

1.[2p]
Put the lines of the proof that $\frac{2 n + 3}{n + 1} \to 2$ in order.

2.[2p]
For $a_{n} = \frac{3 n + 1}{n + 2}$ and $L = 3$ , what is the smallest $N$ such that $| a_{n} - 3 | < 0.01$ for every $n \geq N$ ?

3.[1p]
For $a_{n} = \frac{1}{n}$ and $L = 0$ , what is the smallest $N$ such that $| a_{n} | < 0.003$ for every $n \geq N$ ?

4.[1p]
In the proof that $\frac{1}{n} \to 0$ , which step needs the Archimedean property?

5.[2p]
A proof that $\frac{3 n + 1}{n + 2} \to 3$ says "take $N = \frac{5}{ε}$ ". What would a careful reader object to?

6.[2p]
Which of these are correct about proving that $(- 1)^{n}$ diverges?

7.[3p]
Match each theorem to the move its proof turns on.

8.[1p]
Every bounded sequence of real numbers converges.

9.[3p]
Which of these are true of the decimal truncations $1, 1.4, 1.41, 1.414, \dots$ of $\sqrt{2}$ ?

10.[3p]
Which of these conclusions follow from the algebra of limits, given only that $a_{n} \to 2$ and $b_{n} \to 0$ ?

11.[2p]
To prove $\frac{(- 1)^{n}}{n} \to 0$ by the squeeze theorem, which pair of bounding sequences works?

12.[1p]
Every bounded sequence of real numbers has a convergent subsequence.

13.[2p]
In proving that $f (x) = 3 x - 1$ is continuous at $a = 2$ , what is the largest $δ$ that works for $ε = 0.3$ ?

14.[2p]
Why is $H (x) = 0$ for $x < 0$ , $H (x) = 1$ for $x \geq 0$ not continuous at $0$ ?