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I am probably going a little off-topic with these comments, so feel free to downvote :)

In my opinion this type of proof emphasizes why it is wrong to teach/take “Calculus” instead of Analysis.

For most of the nice applications of integration, we always use the following approach: take some quantity/expression, break it in many pieces, identify the sum of many pieces as a Riemann sum, and thus our quantity is the limit of the Riemann sums, thus the corresponding integral…

Unfortunately, except in serious Analysis courses, not even once do we go into the subtle details: why is the Riemann sum a good approximation for our quantity, namely why does the error in our approximation go to zero.

Most students who take Calculus end up “understanding” lots of false results, which we don’t have the time to disprove in general: any derivative is continuous, any approximation that looks good is good, …

To come back to this problem, not all approximations that look good are good. We always MUST prove that the errors in our approximations go to zero. And for all the formulas we “prove” in calculus, there is an actual mathematical proof, which is pretty technical (and most non-mathematicians would say boring and stupid, but then without such proofs one cannot really understand why the “proof” from the above picture is wrong). But without going through the formal proofs, one cannot truly understand why that particular approximation works in that case, and more importantly why a different approximation won’t work.

Coming back to the above picture, one way to understand it is the following: we approximate the circle by a sequence of polygons. Let $c_n$ be the length of the $n$th polygon and $c$ be the length of the circle. At each step the error in our approximation is $4-\pi$, which doesn’t go to zero. This means that the arclength of the circle might not be the limit of arclengths of the polygons. The only thing we can conclude is that, if all the quantities and limits that appear in the picture exist, then the limit approximates the arclength of the circle with an error of at most the limsup of the errors. In other words, $4 \approx \pi$ with an error less than or equal to $4-\pi$. Hmm, what is wrong with this?

I am probably going a little off-topic with these comments, so feel free to downvote :)

In my opinion this type of proof emphasizes why it is wrong to teach/take “Calculus” instead of Analysis.

For most of the nice applications of integration, we always use the following approach: take some quantity/expression, break it in many pieces, identify the sum of many pieces as a Riemann sum, and thus our quantity is the limit of the Riemann sums, thus the corresponding integral…

Unfortunately, except in serious Analysis courses, not even once do we go into the subtle details: why is the Riemann sum a good approximation for our quantity, namely why does the error in our approximation go to zero…

Most students who take Calculus end up “understanding” lots of false results, which we don’t have the time to disprove in general: any derivative is continuous, any approximation that looks good is good, …

To come back to this problem, not all approximations that look good are good. We always MUST prove that the errors in our approximations go to zero. And for all the formulas we “prove” in calculus, there is an actual mathematical proof, which is pretty technical (and most non-mathematicians would say boring and stupid, but then without such proofs one cannot really understand why the “proof” from the above picture is wrong). But without going through the formal proofs, one cannot truly understand why that particular approximation works in that case, and more importantly why a different approximation won’t work.

Coming back to the above picture, one way to understand it is the following: we approximate the circle by a sequence of polygons. Let $c_n$ be the length of the $n$th polygon and $c$ be the length of the circle. At each step the error in our approximation is $4-\pi$, which doesn’t go to zero. This means that the arclength of the circle might not be the limit of arclengths of the polygons. The only thing we can conclude is that, if all the quantities and limits that appear in the picture exist, then the limit approximates the arclength of the circle with an error of at most the limsup of the errors. In other words, $4 \approx \pi$ with an error less than or equal to $4-\pi$. Hmm, what is wrong with this?

I am probably going a little off-topic with these comments, so feel free to downvote :)

In my opinion this type of proof emphasizes why it is wrong to teach/take “Calculus” instead of Analysis.

For most of the nice applications of integration, we always use the following approach: take some quantity/expression, break it in many pieces, identify the sum of many pieces as a Riemann sum, and thus our quantity is the limit of the Riemann sums, thus the corresponding integral…

Unfortunately, except in serious Analysis courses, not even once do we go into the subtle details: why is the Riemann sum a good approximation for our quantity, namely why does the error in our approximation go to zero…

Most students who take Calculus end up “understanding” lots of false results, which we don’t have the time to disprove in general: any derivative is continuous, any approximation that looks good is good, …

To come back to this problem, not all approximations that look good are good. We always MUST prove that the errors in our approximations go to zero. And for all the formulas we “prove” in calculus, there is an actual mathematical proof, which is pretty technical (and most non-mathematicians would say boring and stupid, but then without such proofs one cannot really understand why the “proof” from the above picture is wrong). But without going through the formal proofs, one cannot truly understand why that particular approximation works in that case, and more importantly why a different approximation won’t work.

Coming back to the above picture, one way to understand it is the following: we approximate the circle by a sequence of polygons. Let $c_n$ be the length of the $n$th polygon and $c$ be the length of the circle. At each step the error in our approximation is $4-\pi$, which doesn’t go to zero. This means that the arclength of the circle might not be the limit of arclengths of the polygons. The only thing we can conclude is that, if all the quantities and limits that appear in the picture exist, then the limit approximates the arclength of the circle with an error of at most the limsup of the errors. In other words, $4 \approx \pi$ with an error less than or equal to $4-\pi$. Hmm, what is wrong with this?

I am probably going a little off-topic with these comments, so feel free to downvote :)

In my opinion this type of proof emphasizes why it is wrong to teach/take “Calculus” instead of Analysis.

For most of the nice applications of integration, we always use the following approach: take some quantity/expression, break it in many pieces, identify the sum of many pieces as a Riemann sum, and thus our quantity is the limit of the Riemann sums, thus the corresponding integral…

Unfortunately, except in serious Analysis courses, not even once do we go into the subtle details: why is the Riemann sum a good approximation for our quantity, namely why does the error in our approximation go to zero.

Most students who take Calculus end up “understanding” lots of false results, which we don’t have the time to disprove in general: any derivative is continuous, any approximation that looks good is good, …

To come back to this problem, not all approximations that look good are good. We always MUST prove that the errors in our approximations go to zero. And for all the formulas we “prove” in calculus, there is an actual mathematical proof, which is pretty technical (and most non-mathematicians would say boring and stupid, but then without such proofs one cannot really understand why the “proof” from the above picture is wrong). But without going through the formal proofs, one cannot truly understand why that particular approximation works in that case, and more importantly why a different approximation won’t work.

Coming back to the above picture, one way to understand it is the following: we approximate the circle by a sequence of polygons. Let $c_n$ be the length of the $n$th polygon and $c$ be the length of the circle. At each step the error in our approximation is $4-\pi$, which doesn’t go to zero. This means that the arclength of the circle might not be the limit of arclengths of the polygons. The only thing we can conclude is that, if all the quantities and limits that appear in the picture exist, then the limit approximates the arclength of the circle with an error of at most the limsup of the errors. In other words, $4 \approx \pi$ with an error less than or equal to $4-\pi$. Hmm, what is wrong with this?

I am probably going a little off topic with these comments, so feel free to downvote :)

In my oppinion this type of proof emphasized why it is wrong to teach/take "Calculus" instead of Analysis.

For most of the nice applications of integration, we always use the following approach: take some quantity/expression; break it in many pieces, identify the sum of many pieces as a Riemmann sum, and thus our quantity is the limit of the Riemann sums, thus the corresponding integral...

Unfortunatelly, excepting in the serious Analysis courses, not even once we go into the suttle details: why is the Riemann sum a good approximation for our quantity, namely why does the error in our approximation go to zero...

Most students which take Calculus end up "understanding" lots of false results, which we don't have the time to disprove in general: any derivative is continuous, any approximation which looks good is good,....

To come back to this problem, not any approximation which looks good is good, we always MUST prove that the error in our approximations goes to zero. And for all the formulas we "prove" in calculus, there is an actual mathematical proof, which is pretty technical (and most non-mathematicians would say boring and stupid, but then without such proofs one cannot really understand why the "proof" from the above picture is wrong). But without going through the formal proofs, one cannot trully understand why that particular approximation works in that case, and more importantly why a different approximation won't work....

Coming back to the above picture, one way to understand it is the following: we approximate the circle by a sequence of polygons. Let $c_n$ be the length of the $n$th polygon and $c$ be the lenght of the circle. At each step the error in our approximation is $4-\pi$, which doesn't go to zero. This means that the arclenght of the circle might not be the limit of arclenghts of the polygons... The only thing we can conclude is that, if all the quantities and limits which appear in the picture exists, then the limit approximates the arclenght of the circle with an error of at most the limsup of the errors. In other words, $4 \sim \pi$ with an error less or equal than $4-\pi$, hmm what is wrong with this?

I am probably going a little off-topic with these comments, so feel free to downvote :)

In my opinion this type of proof emphasizes why it is wrong to teach/take “Calculus” instead of Analysis.

For most of the nice applications of integration, we always use the following approach: take some quantity/expression, break it in many pieces, identify the sum of many pieces as a Riemann sum, and thus our quantity is the limit of the Riemann sums, thus the corresponding integral…

Unfortunately, except in serious Analysis courses, not even once do we go into the subtle details: why is the Riemann sum a good approximation for our quantity, namely why does the error in our approximation go to zero…

Most students who take Calculus end up “understanding” lots of false results, which we don’t have the time to disprove in general: any derivative is continuous, any approximation that looks good is good, …

To come back to this problem, not all approximations that look good are good. We always MUST prove that the errors in our approximations go to zero. And for all the formulas we “prove” in calculus, there is an actual mathematical proof, which is pretty technical (and most non-mathematicians would say boring and stupid, but then without such proofs one cannot really understand why the “proof” from the above picture is wrong). But without going through the formal proofs, one cannot truly understand why that particular approximation works in that case, and more importantly why a different approximation won’t work.

Coming back to the above picture, one way to understand it is the following: we approximate the circle by a sequence of polygons. Let $c_n$ be the length of the $n$th polygon and $c$ be the length of the circle. At each step the error in our approximation is $4-\pi$, which doesn’t go to zero. This means that the arclength of the circle might not be the limit of arclengths of the polygons. The only thing we can conclude is that, if all the quantities and limits that appear in the picture exist, then the limit approximates the arclength of the circle with an error of at most the limsup of the errors. In other words, $4 \approx \pi$ with an error less than or equal to $4-\pi$. Hmm, what is wrong with this?