Proving that between any two real numbers there exist a real number
Formally, I want to prove that if $x$ and $y$ are real numbers such that $x lt y$ then there exists a real number $z$ such that $x lt z lt y$.
Is this equivalent to proving the real line is continuous?
real-numbers
asked 54 mins ago
MHall
707
add a comment |
Formally, I want to prove that if $x$ and $y$ are real numbers such that $x lt y$ then there exists a real number $z$ such that $x lt z lt y$.
Is this equivalent to proving the real line is continuous?
real-numbers
asked 54 mins ago
MHall
707
Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x?
– user247327
50 mins ago
1
By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $Bbb R$ is a complete metric space.
– ÍgjøgnumMeg
48 mins ago
add a comment |
Formally, I want to prove that if $x$ and $y$ are real numbers such that $x lt y$ then there exists a real number $z$ such that $x lt z lt y$.
Is this equivalent to proving the real line is continuous?
real-numbers
asked 54 mins ago
MHall
707
Formally, I want to prove that if $x$ and $y$ are real numbers such that $x lt y$ then there exists a real number $z$ such that $x lt z lt y$.
Is this equivalent to proving the real line is continuous?
real-numbers
real-numbers
asked 54 mins ago
MHall
707
asked 54 mins ago
MHall
707
asked 54 mins ago
MHall
707
asked 54 mins ago
MHall
707
asked 54 mins ago
MHall
707
707
Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x?
– user247327
50 mins ago
1
By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $Bbb R$ is a complete metric space.
– ÍgjøgnumMeg
48 mins ago
add a comment |
Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x?
– user247327
50 mins ago
1
By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $Bbb R$ is a complete metric space.
– ÍgjøgnumMeg
48 mins ago
Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x?
– user247327
50 mins ago
Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x?
– user247327
50 mins ago
1
1
By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $Bbb R$ is a complete metric space.
– ÍgjøgnumMeg
48 mins ago
By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $Bbb R$ is a complete metric space.
– ÍgjøgnumMeg
48 mins ago
add a comment |
1 Answer
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First, let me say a bit about the end of your question. I'm not sure what you mean by "the real line is continuous," but I suspect you mean that the real line is connected or complete. If so, then no, that's a different thing; for example, $mathbb{Q}$ has the "density" property here (between any two rationals, there's another rational) but is neither connected (consider $(-infty,pi)$ versus $(pi,infty)$, for example) or complete (consider the sequence $3,3.1,3.14,3.141,...$, for example).
Since connectedness/completeness can be a bit technical at first, let me say the following: intuitively, a space is connected if it doesn't "break into a bunch of separate pieces," and is complete if it "doesn't have any gaps" - the set $mathbb{Q}$ of rationals, even though it's dense, has lots of gaps and breaks apart really easily.
As to proving the claim you're looking at, I suspect you're thinking too hard about it. First, try to guess what a good formula for such a $z$ might be, and then try to prove (using whatever formal system you've been given) that it actually works. And this formula will be quite simple.
HINT: if on a quiz Sam scored $10/10$ and Alex scored $6/10$, then together they got an ---- score of $8/10$. Now, how do you calculate the "---" of two numbers?
answered 50 mins ago
Noah Schweber
121k10147282
add a comment |
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First, let me say a bit about the end of your question. I'm not sure what you mean by "the real line is continuous," but I suspect you mean that the real line is connected or complete. If so, then no, that's a different thing; for example, $mathbb{Q}$ has the "density" property here (between any two rationals, there's another rational) but is neither connected (consider $(-infty,pi)$ versus $(pi,infty)$, for example) or complete (consider the sequence $3,3.1,3.14,3.141,...$, for example).
Since connectedness/completeness can be a bit technical at first, let me say the following: intuitively, a space is connected if it doesn't "break into a bunch of separate pieces," and is complete if it "doesn't have any gaps" - the set $mathbb{Q}$ of rationals, even though it's dense, has lots of gaps and breaks apart really easily.
As to proving the claim you're looking at, I suspect you're thinking too hard about it. First, try to guess what a good formula for such a $z$ might be, and then try to prove (using whatever formal system you've been given) that it actually works. And this formula will be quite simple.
HINT: if on a quiz Sam scored $10/10$ and Alex scored $6/10$, then together they got an ---- score of $8/10$. Now, how do you calculate the "---" of two numbers?
answered 50 mins ago
Noah Schweber
121k10147282
add a comment |
First, let me say a bit about the end of your question. I'm not sure what you mean by "the real line is continuous," but I suspect you mean that the real line is connected or complete. If so, then no, that's a different thing; for example, $mathbb{Q}$ has the "density" property here (between any two rationals, there's another rational) but is neither connected (consider $(-infty,pi)$ versus $(pi,infty)$, for example) or complete (consider the sequence $3,3.1,3.14,3.141,...$, for example).
Since connectedness/completeness can be a bit technical at first, let me say the following: intuitively, a space is connected if it doesn't "break into a bunch of separate pieces," and is complete if it "doesn't have any gaps" - the set $mathbb{Q}$ of rationals, even though it's dense, has lots of gaps and breaks apart really easily.
As to proving the claim you're looking at, I suspect you're thinking too hard about it. First, try to guess what a good formula for such a $z$ might be, and then try to prove (using whatever formal system you've been given) that it actually works. And this formula will be quite simple.
HINT: if on a quiz Sam scored $10/10$ and Alex scored $6/10$, then together they got an ---- score of $8/10$. Now, how do you calculate the "---" of two numbers?
answered 50 mins ago
Noah Schweber
121k10147282
add a comment |
First, let me say a bit about the end of your question. I'm not sure what you mean by "the real line is continuous," but I suspect you mean that the real line is connected or complete. If so, then no, that's a different thing; for example, $mathbb{Q}$ has the "density" property here (between any two rationals, there's another rational) but is neither connected (consider $(-infty,pi)$ versus $(pi,infty)$, for example) or complete (consider the sequence $3,3.1,3.14,3.141,...$, for example).
Since connectedness/completeness can be a bit technical at first, let me say the following: intuitively, a space is connected if it doesn't "break into a bunch of separate pieces," and is complete if it "doesn't have any gaps" - the set $mathbb{Q}$ of rationals, even though it's dense, has lots of gaps and breaks apart really easily.
As to proving the claim you're looking at, I suspect you're thinking too hard about it. First, try to guess what a good formula for such a $z$ might be, and then try to prove (using whatever formal system you've been given) that it actually works. And this formula will be quite simple.
HINT: if on a quiz Sam scored $10/10$ and Alex scored $6/10$, then together they got an ---- score of $8/10$. Now, how do you calculate the "---" of two numbers?
answered 50 mins ago
Noah Schweber
121k10147282
First, let me say a bit about the end of your question. I'm not sure what you mean by "the real line is continuous," but I suspect you mean that the real line is connected or complete. If so, then no, that's a different thing; for example, $mathbb{Q}$ has the "density" property here (between any two rationals, there's another rational) but is neither connected (consider $(-infty,pi)$ versus $(pi,infty)$, for example) or complete (consider the sequence $3,3.1,3.14,3.141,...$, for example).
Since connectedness/completeness can be a bit technical at first, let me say the following: intuitively, a space is connected if it doesn't "break into a bunch of separate pieces," and is complete if it "doesn't have any gaps" - the set $mathbb{Q}$ of rationals, even though it's dense, has lots of gaps and breaks apart really easily.
As to proving the claim you're looking at, I suspect you're thinking too hard about it. First, try to guess what a good formula for such a $z$ might be, and then try to prove (using whatever formal system you've been given) that it actually works. And this formula will be quite simple.
HINT: if on a quiz Sam scored $10/10$ and Alex scored $6/10$, then together they got an ---- score of $8/10$. Now, how do you calculate the "---" of two numbers?
answered 50 mins ago
Noah Schweber
121k10147282
answered 50 mins ago
Noah Schweber
121k10147282
answered 50 mins ago
Noah Schweber
121k10147282
answered 50 mins ago
Noah Schweber
121k10147282
121k10147282
add a comment |
add a comment |
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Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x?
– user247327
50 mins ago
1
By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $Bbb R$ is a complete metric space.
– ÍgjøgnumMeg
48 mins ago