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 $N > \frac{1}{\varepsilon}$

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

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

Letting $ε$ be an arbitrary positive number

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

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

$N$ must be the smallest position that works

The answer is: $\frac{5}{\varepsilon}$ 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: $\frac{5}{\varepsilon}$ 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 $\varepsilon = 1$, For each $N$, one of the terms at positions $2N$ and $2N + 1$ 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

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?

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 $\sqrt{2}$ ?