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What came first, sequence or $epsilon-delta$ criterium for limit?
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I find in many places that first came the sequence criterion, on Cauchy's book "Cours d'analyse" but I am not sure. Does any body know anything about that?
Thank you!
real-analysis math-history
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
asked Jul 24 at 3:04
John
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up vote
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I find in many places that first came the sequence criterion, on Cauchy's book "Cours d'analyse" but I am not sure. Does any body know anything about that?
Thank you!
real-analysis math-history
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
asked Jul 24 at 3:04
John
765
Depends on what you mean by a definition and how rigorous you expect the statement to be with respect to modern understanding. Sequential limits are almost certainly more primitive, and were understood (in some sense) by the Greeks. Consider, for example, Archimedes method of determining $pi$, or the arguments that follow from Zeno's paradoxes. The $varepsilon$-$delta$ formalism is much more recent. It has a late-19th early-20th C vibe to me, and probably doesn't predate the formalism of a "function", which (to my recollection) is largely due to Fourier (late-18th, early-19th C).
– Xander Henderson
Jul 24 at 3:12
What exactly do you mean by sequence criteria? Do you mean $epsilon$-$N$ vs. $epsilon$-$delta$?
– anakhro
Jul 24 at 4:12
@Xander so according to what you are saying, the definition with sequences came first?
– John
Jul 24 at 4:20
@anakhronizein exactly, that $lim_xto af(x)=L$ iff, for every sequence $(x_n)in D_fsetminus a$, st $x_nto a$, implies that $f(x_n)to L$.
– John
Jul 24 at 4:22
2
I'm voting to close this question as off-topic because it may well belong on the History of Science and Mathematics SE.
– Parcly Taxel
Jul 24 at 16:43
 |Â
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I find in many places that first came the sequence criterion, on Cauchy's book "Cours d'analyse" but I am not sure. Does any body know anything about that?
Thank you!
real-analysis math-history
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
asked Jul 24 at 3:04
John
765
I find in many places that first came the sequence criterion, on Cauchy's book "Cours d'analyse" but I am not sure. Does any body know anything about that?
Thank you!
real-analysis math-history
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
asked Jul 24 at 3:04
John
765
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
edited Jul 24 at 15:48
David C. Ullrich
54.1k33481
54.1k33481
asked Jul 24 at 3:04
John
765
asked Jul 24 at 3:04
John
765
asked Jul 24 at 3:04
John
765
765
Depends on what you mean by a definition and how rigorous you expect the statement to be with respect to modern understanding. Sequential limits are almost certainly more primitive, and were understood (in some sense) by the Greeks. Consider, for example, Archimedes method of determining $pi$, or the arguments that follow from Zeno's paradoxes. The $varepsilon$-$delta$ formalism is much more recent. It has a late-19th early-20th C vibe to me, and probably doesn't predate the formalism of a "function", which (to my recollection) is largely due to Fourier (late-18th, early-19th C).
– Xander Henderson
Jul 24 at 3:12
What exactly do you mean by sequence criteria? Do you mean $epsilon$-$N$ vs. $epsilon$-$delta$?
– anakhro
Jul 24 at 4:12
@Xander so according to what you are saying, the definition with sequences came first?
– John
Jul 24 at 4:20
@anakhronizein exactly, that $lim_xto af(x)=L$ iff, for every sequence $(x_n)in D_fsetminus a$, st $x_nto a$, implies that $f(x_n)to L$.
– John
Jul 24 at 4:22
2
I'm voting to close this question as off-topic because it may well belong on the History of Science and Mathematics SE.
– Parcly Taxel
Jul 24 at 16:43
 |Â
show 1 more comment
Depends on what you mean by a definition and how rigorous you expect the statement to be with respect to modern understanding. Sequential limits are almost certainly more primitive, and were understood (in some sense) by the Greeks. Consider, for example, Archimedes method of determining $pi$, or the arguments that follow from Zeno's paradoxes. The $varepsilon$-$delta$ formalism is much more recent. It has a late-19th early-20th C vibe to me, and probably doesn't predate the formalism of a "function", which (to my recollection) is largely due to Fourier (late-18th, early-19th C).
– Xander Henderson
Jul 24 at 3:12
What exactly do you mean by sequence criteria? Do you mean $epsilon$-$N$ vs. $epsilon$-$delta$?
– anakhro
Jul 24 at 4:12
@Xander so according to what you are saying, the definition with sequences came first?
– John
Jul 24 at 4:20
@anakhronizein exactly, that $lim_xto af(x)=L$ iff, for every sequence $(x_n)in D_fsetminus a$, st $x_nto a$, implies that $f(x_n)to L$.
– John
Jul 24 at 4:22
2
I'm voting to close this question as off-topic because it may well belong on the History of Science and Mathematics SE.
– Parcly Taxel
Jul 24 at 16:43
Depends on what you mean by a definition and how rigorous you expect the statement to be with respect to modern understanding. Sequential limits are almost certainly more primitive, and were understood (in some sense) by the Greeks. Consider, for example, Archimedes method of determining $pi$, or the arguments that follow from Zeno's paradoxes. The $varepsilon$-$delta$ formalism is much more recent. It has a late-19th early-20th C vibe to me, and probably doesn't predate the formalism of a "function", which (to my recollection) is largely due to Fourier (late-18th, early-19th C).
– Xander Henderson
Jul 24 at 3:12
Depends on what you mean by a definition and how rigorous you expect the statement to be with respect to modern understanding. Sequential limits are almost certainly more primitive, and were understood (in some sense) by the Greeks. Consider, for example, Archimedes method of determining $pi$, or the arguments that follow from Zeno's paradoxes. The $varepsilon$-$delta$ formalism is much more recent. It has a late-19th early-20th C vibe to me, and probably doesn't predate the formalism of a "function", which (to my recollection) is largely due to Fourier (late-18th, early-19th C).
– Xander Henderson
Jul 24 at 3:12
What exactly do you mean by sequence criteria? Do you mean $epsilon$-$N$ vs. $epsilon$-$delta$?
– anakhro
Jul 24 at 4:12
What exactly do you mean by sequence criteria? Do you mean $epsilon$-$N$ vs. $epsilon$-$delta$?
– anakhro
Jul 24 at 4:12
@Xander so according to what you are saying, the definition with sequences came first?
– John
Jul 24 at 4:20
@Xander so according to what you are saying, the definition with sequences came first?
– John
Jul 24 at 4:20
@anakhronizein exactly, that $lim_xto af(x)=L$ iff, for every sequence $(x_n)in D_fsetminus a$, st $x_nto a$, implies that $f(x_n)to L$.
– John
Jul 24 at 4:22
@anakhronizein exactly, that $lim_xto af(x)=L$ iff, for every sequence $(x_n)in D_fsetminus a$, st $x_nto a$, implies that $f(x_n)to L$.
– John
Jul 24 at 4:22
2
2
I'm voting to close this question as off-topic because it may well belong on the History of Science and Mathematics SE.
– Parcly Taxel
Jul 24 at 16:43
I'm voting to close this question as off-topic because it may well belong on the History of Science and Mathematics SE.
– Parcly Taxel
Jul 24 at 16:43
 |Â
show 1 more comment
1 Answer
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The $varepsilon$-$delta$ formulation of limits (Wikipedia) was anticipated faintly by Fermat, used implicitly by Cauchy, and formalized by Bolzano in 1817. As Xander Henderson indicated in the comments, the notion of a sequence limit (Wikipedia again)—as $n to infty$, for instance—is older: anticipated by Archimedes in his method of exhaustion. However, its formalism dates also to Bolzano (in 1816), but it was apparently not much noticed, and it was reformulated by Weierstrass in the 1870s.
See the "History" section of either Wikipedia article for more details.
answered Jul 24 at 5:42
Brian Tung
25.2k32453
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The $varepsilon$-$delta$ formulation of limits (Wikipedia) was anticipated faintly by Fermat, used implicitly by Cauchy, and formalized by Bolzano in 1817. As Xander Henderson indicated in the comments, the notion of a sequence limit (Wikipedia again)—as $n to infty$, for instance—is older: anticipated by Archimedes in his method of exhaustion. However, its formalism dates also to Bolzano (in 1816), but it was apparently not much noticed, and it was reformulated by Weierstrass in the 1870s.
See the "History" section of either Wikipedia article for more details.
answered Jul 24 at 5:42
Brian Tung
25.2k32453
add a comment |Â
up vote
0
down vote
The $varepsilon$-$delta$ formulation of limits (Wikipedia) was anticipated faintly by Fermat, used implicitly by Cauchy, and formalized by Bolzano in 1817. As Xander Henderson indicated in the comments, the notion of a sequence limit (Wikipedia again)—as $n to infty$, for instance—is older: anticipated by Archimedes in his method of exhaustion. However, its formalism dates also to Bolzano (in 1816), but it was apparently not much noticed, and it was reformulated by Weierstrass in the 1870s.
See the "History" section of either Wikipedia article for more details.
answered Jul 24 at 5:42
Brian Tung
25.2k32453
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The $varepsilon$-$delta$ formulation of limits (Wikipedia) was anticipated faintly by Fermat, used implicitly by Cauchy, and formalized by Bolzano in 1817. As Xander Henderson indicated in the comments, the notion of a sequence limit (Wikipedia again)—as $n to infty$, for instance—is older: anticipated by Archimedes in his method of exhaustion. However, its formalism dates also to Bolzano (in 1816), but it was apparently not much noticed, and it was reformulated by Weierstrass in the 1870s.
See the "History" section of either Wikipedia article for more details.
answered Jul 24 at 5:42
Brian Tung
25.2k32453
The $varepsilon$-$delta$ formulation of limits (Wikipedia) was anticipated faintly by Fermat, used implicitly by Cauchy, and formalized by Bolzano in 1817. As Xander Henderson indicated in the comments, the notion of a sequence limit (Wikipedia again)—as $n to infty$, for instance—is older: anticipated by Archimedes in his method of exhaustion. However, its formalism dates also to Bolzano (in 1816), but it was apparently not much noticed, and it was reformulated by Weierstrass in the 1870s.
See the "History" section of either Wikipedia article for more details.
answered Jul 24 at 5:42
Brian Tung
25.2k32453
edited Jul 26 at 4:54
answered Jul 24 at 5:42
Brian Tung
25.2k32453
answered Jul 24 at 5:42
Brian Tung
25.2k32453
answered Jul 24 at 5:42
Brian Tung
25.2k32453
25.2k32453
add a comment |Â
add a comment |Â
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Depends on what you mean by a definition and how rigorous you expect the statement to be with respect to modern understanding. Sequential limits are almost certainly more primitive, and were understood (in some sense) by the Greeks. Consider, for example, Archimedes method of determining $pi$, or the arguments that follow from Zeno's paradoxes. The $varepsilon$-$delta$ formalism is much more recent. It has a late-19th early-20th C vibe to me, and probably doesn't predate the formalism of a "function", which (to my recollection) is largely due to Fourier (late-18th, early-19th C).
– Xander Henderson
Jul 24 at 3:12
What exactly do you mean by sequence criteria? Do you mean $epsilon$-$N$ vs. $epsilon$-$delta$?
– anakhro
Jul 24 at 4:12
@Xander so according to what you are saying, the definition with sequences came first?
– John
Jul 24 at 4:20
@anakhronizein exactly, that $lim_xto af(x)=L$ iff, for every sequence $(x_n)in D_fsetminus a$, st $x_nto a$, implies that $f(x_n)to L$.
– John
Jul 24 at 4:22
2
I'm voting to close this question as off-topic because it may well belong on the History of Science and Mathematics SE.
– Parcly Taxel
Jul 24 at 16:43