Lesson test: The real numbers

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The real numbers

1.[1p]
Which property of $ℝ$ fails in $ℚ$ ?

Every nonempty set bounded above has a least upper bound

Correct

Every nonzero element has a reciprocal

The answer is: Every nonempty set bounded above has a least upper bound

The order is total

The answer is: Every nonempty set bounded above has a least upper bound

$x \leq y$ implies $x + z \leq y + z$

The answer is: Every nonempty set bounded above has a least upper bound

2.[1p]
The ordered field axioms alone are enough to prove that some number has square $2$ .

True

The answer is: False

False

Correct

3.[2p]
Let $s = \sup A$ and $ε = 0.01$ . What does the characterisation of the supremum guarantee?

Some $a \in A$ satisfies $a > s - 0.01$

Correct

Some $a \in A$ satisfies $a = s - 0.01$

The answer is: Some $a \in A$ satisfies $a > s - 0.01$

Every $a \in A$ satisfies $a > s - 0.01$

The answer is: Some $a \in A$ satisfies $a > s - 0.01$

$s - 0.01$ is an upper bound of $A$

The answer is: Some $a \in A$ satisfies $a > s - 0.01$

4.[2p]
By the Archimedean property some $n \in ℕ$ has $\frac{1}{n} < 0.03$ . What is the smallest such $n$ ?

CorrectNot quite: 34

5.[2p]
What is $\sup {x \in ℝ : x^{2} + x < 6}$ ?

CorrectNot quite: 2

6.[2p]
Put the lines of the proof of the Archimedean property in order.

Since $ℕ$ is nonempty, completeness gives it a supremum $s$ .
Since $s - 1$ is not an upper bound, some $n \in ℕ$ has $n > s - 1$ .
This contradicts that $s$ is an upper bound of $ℕ$ .
Then $n + 1 > s$ and $n + 1 \in ℕ$ .
Suppose, for contradiction, that $ℕ$ is bounded above in $ℝ$ .

Show the answer

a, b, c, d, e

7.[3p]
Match each result to the step its proof turns on.

No least upper bound of ${ q \in ℚ
Archimedean property
Density of $ℚ$ in $ℝ$
Existence of $\sqrt{2}$

the least integer $m$ greater than $n a$
the supremum of ${x \in ℝ : x^{2} < 2}$ has neither $s^{2} < 2$ nor $s^{2} > 2$
q^2 < 2 } $i n$ ℚ $: t h e i r r a t i o n a l i t y o f$ \sqrt{2} $r u l e s o u t$ u^2 = 2$
a supremum of $ℕ$ would be exceeded by $n + 1$

Show the answer

No least upper bound of ${q \in ℚ : q^{2} < 2}$ in $ℚ$ : the irrationality of $\sqrt{2}$ rules out $u^{2} = 2$ Archimedean property: a supremum of $ℕ$ would be exceeded by $n + 1$ Density of $ℚ$ in $ℝ$ : the least integer $m$ greater than $n a$ Existence of $\sqrt{2}$ : the supremum of ${x \in ℝ : x^{2} < 2}$ has neither $s^{2} < 2$ nor $s^{2} > 2$

8.[3p]
Which statements are true?

Select all that apply

$\inf {\frac{1}{n} : n \in ℕ} = 0$

Correct

Between any two distinct reals lies a rational

Correct

Between any two distinct reals lies an irrational

Correct

Every nonempty set of reals bounded below has an infimum

Correct

Every nonempty set of rationals bounded above has a least upper bound in $ℚ$

The answer is: $\inf \{ \frac{1}{n} : n \in ℕ \} = 0$, Between any two distinct reals lies a rational, Between any two distinct reals lies an irrational, Every nonempty set of reals bounded below has an infimum

$ℕ$ has a supremum in $ℝ$

9.[2p]
In the proof that $s = \sup {x \in ℝ : x^{2} < 2}$ has $s^{2} = 2$ , why can $s^{2} > 2$ not hold?

Then $s - k$ , with $k = \frac{s^{2} - 2}{2 s}$ , would be a smaller upper bound

Correct

Then $s + h$ would lie in the set and exceed $s$

The answer is: Then $s - k$, with $k = \frac{s^2 - 2}{2s}$, would be a smaller upper bound

Then $s$ would be rational, contradicting the irrationality of $\sqrt{2}$

The answer is: Then $s - k$, with $k = \frac{s^2 - 2}{2s}$, would be a smaller upper bound

Then the set would have no upper bound at all

The answer is: Then $s - k$, with $k = \frac{s^2 - 2}{2s}$, would be a smaller upper bound

1.[1p] Which property of ℝ fails in ℚ?

2.[1p] The ordered field axioms alone are enough to prove that some number has square 2.

3.[2p] Let s=supA and ε=0.01. What does the characterisation of the supremum guarantee?

4.[2p] By the Archimedean property some n∈ℕ has 1n<0.03. What is the smallest such n?

5.[2p] What is sup{x∈ℝ:x2+x<6}?

6.[2p] Put the lines of the proof of the Archimedean property in order.

7.[3p] Match each result to the step its proof turns on.

8.[3p] Which statements are true?

9.[2p] In the proof that s=sup{x∈ℝ:x2<2} has s2=2, why can s2>2 not hold?

1.[1p]
Which property of $ℝ$ fails in $ℚ$ ?

2.[1p]
The ordered field axioms alone are enough to prove that some number has square $2$ .

3.[2p]
Let $s = \sup A$ and $ε = 0.01$ . What does the characterisation of the supremum guarantee?

4.[2p]
By the Archimedean property some $n \in ℕ$ has $\frac{1}{n} < 0.03$ . What is the smallest such $n$ ?

5.[2p]
What is $\sup {x \in ℝ : x^{2} + x < 6}$ ?

6.[2p]
Put the lines of the proof of the Archimedean property in order.

7.[3p]
Match each result to the step its proof turns on.

8.[3p]
Which statements are true?

9.[2p]
In the proof that $s = \sup {x \in ℝ : x^{2} < 2}$ has $s^{2} = 2$ , why can $s^{2} > 2$ not hold?