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Here are a few possible leads.

• Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: $$0.999999... = 1$$
• Given two real numbers $a$ and $b$, consider the three quantities $d_☾(a,b)$, $d_☆(a, b)$ and $d_🪐(a, b)$ defined by: \begin{align*}d_☾(a, b) & = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\ d_☆(a, b) & = \inf\left\{\varepsilon > 0 \,\mid\,\left|{a-b}\right|<\varepsilon \right\} \\ d_🪐(a, b) & = \left|{a-b}\right| \end{align*} then have your students prove that: $$d_☾ = d_☆ = d_🪐$$ then reword your previous theorem as:$$\forall a, b \in \mathbb{R},\quad d_☆(a,b) = 0 \iff a = b$$
• Consider the definition of convergence for a sequence $(b_n)_{n \in \mathbb{N}}$ converging to a limit $a$: $$\forall \varepsilon > 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| < \varepsilon$$ then discuss the specific case of a stationary sequence, i.e the case where you have a number $b$ such that $\forall n \in \mathbb{N},\, b_n = b$.
• Consider the following propositions for two converging sequences $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$: \begin{align*} \forall n,\, a_n \leq b_n & \implies \lim_n a_n \leq \lim_n b_n \\ \forall n,\, a_n < b_n & \implies \lim_n a_n \leq \lim_n b_n \end{align*} And note how we are forced to accept the possibility of equality appearing in the limit.

Note that I used a lot of formal notation in the second and third points, but the notion of "distance between two points" is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion. As for the definition of convergence of a sequence, you can illustrate it graphically, then ask your students what this graph becomes in the case of a constant sequence.

Also, if your students are not familiar with the definition of $\inf$, then great! Rewrite the definition of $d_☆$ using only $\min$ and discuss the special cases. This will force you to make things more explicit and deal with what the abstract $\forall \varepsilon > 0$ really hides.

Here are a few possible leads.

• Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: $$0.999999... = 1$$
• Given two real numbers $a$ and $b$, consider the three quantities $d_☾(a,b)$, $d_☆(a, b)$ and $d_🪐(a, b)$ defined by: \begin{align*}d_☾(a, b) & = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\ d_☆(a, b) & = \inf\left\{\varepsilon > 0 \,\mid\,\left|{a-b}\right|<\varepsilon \right\} \\ d_🪐(a, b) & = \left|{a-b}\right| \end{align*} then have your students prove that: $$d_☾ = d_☆ = d_🪐$$ then reword your previous theorem as:$$\forall a, b \in \mathbb{R},\quad d_☆(a,b) = 0 \iff a = b$$
• Consider the definition of convergence for a sequence $(b_n)_{n \in \mathbb{N}}$ converging to a limit $a$: $$\forall \varepsilon > 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| < \varepsilon$$ then discuss the specific case of a stationary sequence, i.e the case where you have a number $b$ such that $\forall n \in \mathbb{N},\, b_n = b$.
• Consider the following propositions for two converging sequences $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$: \begin{align*} \forall n,\, a_n \leq b_n & \implies \lim_n a_n \leq \lim_n b_n \\ \forall n,\, a_n < b_n & \implies \lim_n a_n \leq \lim_n b_n \end{align*} And note how we are forced to accept the possibility of equality appearing in the limit.

Note that I used a lot of formal notation in the second and third points, but the notion of "distance between two points" is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion. As for the definition of convergence of a sequence, you can illustrate it graphically, then ask your students what this graph becomes in the case of a constant sequence.

Also, if your students are not familiar with the definition of $\inf$, then great! Rewrite the definition of $d_☆$ using only $\min$ and discuss the special cases. This will force you to make things more explicit and deal with what the abstract $\forall \varepsilon > 0$ really hides.

There are several possible proofs for the equality $0.999... = 1$; one possible takeaway is "they are equal because you cannot squeeze a third number inbetween", ie there is no $x$ such that $0.999... < x < 1$. Or alternatively they are equal because the distance between them is 0. In fact it's a direct application of your $\forall \varepsilon$-proposition: since $0.9 < 0.999... < 1$, we have $\left| 1 - 0.999... \right| < 0.1$, and likewise $0.99 < 0.999... < 1$, and $0.999 < 0.999... < 1$, etc. with any finite number of 9 after the decimal point, and thus $\left| 1 - 0.999... \right| < 0.0001$ for any number of 0 before the 1, which can be rewritten as $\forall n \in \mathbb{N},\, \left| 1 - 0.999... \right| < 10^{-n}$, which is equivalent to $\forall \varepsilon > 0,\, \left| 1 - 0.999... \right| < \varepsilon$, which by your proposition is equivalent to $0.999... = 1$.

Here are a few possible leads.

• Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: $$0.999999... = 1$$
• Given two real numbers $a$ and $b$, consider the three quantities $d_☾(a,b)$, $d_☆(a, b)$ and $d_🪐(a, b)$ defined by: \begin{align*}d_☾(a, b) & = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\ d_☆(a, b) & = \inf\left\{\varepsilon > 0 \,\mid\,\left|{a-b}\right|<\varepsilon \right\} \\ d_🪐(a, b) & = \left|{a-b}\right| \end{align*} then have your students prove that: $$d_☾ = d_☆ = d_🪐$$ then reword your previous theorem as:$$\forall a, b \in \mathbb{R},\quad d_☆(a,b) = 0 \iff a = b$$
• Consider the definition of convergence for a sequence $(b_n)_{n \in \mathbb{N}}$ converging to a limit $a$: $$\forall \varepsilon > 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| < \varepsilon$$ then discuss the specific case of a stationary sequence, i.e the case where you have a number $b$ such that $\forall n \in \mathbb{N},\, b_n = b$.

Note that I used a lot of formal notation in the second and third points, but the notion of "distance between two points" is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion. As for the definition of convergence of a sequence, you can illustrate it graphically, then ask your students what this graph becomes in the case of a constant sequence.

Also, if your students are not familiar with the definition of $\inf$, then great! Rewrite the definition of $d_☆$ using only $\min$ and discuss the special cases. This will force you to make things more explicit and deal with what the abstract $\forall \varepsilon > 0$ really hides.

Here are a few possible leads.

• Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: $$0.999999... = 1$$
• Given two real numbers $a$ and $b$, consider the three quantities $d_☾(a,b)$, $d_☆(a, b)$ and $d_🪐(a, b)$ defined by: \begin{align*}d_☾(a, b) & = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\ d_☆(a, b) & = \inf\left\{\varepsilon > 0 \,\mid\,\left|{a-b}\right|<\varepsilon \right\} \\ d_🪐(a, b) & = \left|{a-b}\right| \end{align*} then have your students prove that: $$d_☾ = d_☆ = d_🪐$$ then reword your previous theorem as:$$\forall a, b \in \mathbb{R},\quad d_☆(a,b) = 0 \iff a = b$$
• Consider the definition of convergence for a sequence $(b_n)_{n \in \mathbb{N}}$ converging to a limit $a$: $$\forall \varepsilon > 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| < \varepsilon$$ then discuss the specific case of a stationary sequence, i.e the case where you have a number $b$ such that $\forall n \in \mathbb{N},\, b_n = b$.
• Consider the following propositions for two converging sequences $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$: \begin{align*} \forall n,\, a_n \leq b_n & \implies \lim_n a_n \leq \lim_n b_n \\ \forall n,\, a_n < b_n & \implies \lim_n a_n \leq \lim_n b_n \end{align*} And note how we are forced to accept the possibility of equality appearing in the limit.

Note that I used a lot of formal notation in the second and third points, but the notion of "distance between two points" is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion. As for the definition of convergence of a sequence, you can illustrate it graphically, then ask your students what this graph becomes in the case of a constant sequence.

Also, if your students are not familiar with the definition of $\inf$, then great! Rewrite the definition of $d_☆$ using only $\min$ and discuss the special cases. This will force you to make things more explicit and deal with what the abstract $\forall \varepsilon > 0$ really hides.

Here are a few possible leads.

• Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: $$0.999999... = 1$$
• Given two real numbers $a$ and $b$, consider the two quantities $d_\leq(a,b)$, $d_<(a, b)$ and $d_{|.|}(a, b)$ defined by: \begin{align*}d_\leq(a, b) & = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\ d_<(a, b) & = \inf\left\{\varepsilon > 0 \,\mid\,\left|{a-b}\right|<\varepsilon \right\} \\ d_{|.|}(a, b) & = \left|{a-b}\right| \end{align*} then have your students prove that: $$d_\leq = d_< = d_{|.|}$$ then reword your previous theorem as:$$d_<(a,b) = 0 \iff a = b$$
• Consider the definition of convergence for a sequence $(b_n)_{n \in \mathbb{N}}$ converging to a limit $a$: $$\forall \varepsilon > 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| < \varepsilon$$ then discuss the specific case of a stationary sequence, i.e the case where $\forall n \in \mathbb{N},\, b_n = b_0$.

Note that I used a lot of formal notation in the second and third points, but the notion of "distance between two points" is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion.

Also, if your students are not familiar with the definition of $\inf$, then great! Rewrite the definition of $d_<$ using only $\min$ and discuss the special cases. This will force you to make things more explicit and deal with what the abstract $\forall \varepsilon > 0$ really hides.

Here are a few possible leads.

• Ask your students to prove the following equality, then discuss with them whether they find this intuitive or counterintuitive, and why: $$0.999999... = 1$$
• Given two real numbers $a$ and $b$, consider the three quantities $d_☾(a,b)$, $d_☆(a, b)$ and $d_🪐(a, b)$ defined by: \begin{align*}d_☾(a, b) & = \min\left\{\varepsilon \geq 0 \,\mid\,\left|{a-b}\right|\leq\varepsilon \right\} \\ d_☆(a, b) & = \inf\left\{\varepsilon > 0 \,\mid\,\left|{a-b}\right|<\varepsilon \right\} \\ d_🪐(a, b) & = \left|{a-b}\right| \end{align*} then have your students prove that: $$d_☾ = d_☆ = d_🪐$$ then reword your previous theorem as:$$\forall a, b \in \mathbb{R},\quad d_☆(a,b) = 0 \iff a = b$$
• Consider the definition of convergence for a sequence $(b_n)_{n \in \mathbb{N}}$ converging to a limit $a$: $$\forall \varepsilon > 0,\, \exists N \in \mathbb{N},\, \forall n \geq N,\, |a-b_n| < \varepsilon$$ then discuss the specific case of a stationary sequence, i.e the case where you have a number $b$ such that $\forall n \in \mathbb{N},\, b_n = b$.

Note that I used a lot of formal notation in the second and third points, but the notion of "distance between two points" is meant to be rather intuitive and the fact that we have more than one manner to define it with formal symbols should spark a discussion. As for the definition of convergence of a sequence, you can illustrate it graphically, then ask your students what this graph becomes in the case of a constant sequence.

Also, if your students are not familiar with the definition of $\inf$, then great! Rewrite the definition of $d_☆$ using only $\min$ and discuss the special cases. This will force you to make things more explicit and deal with what the abstract $\forall \varepsilon > 0$ really hides.