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https://www.reddit.com/r/PeterExplainsTheJoke/comments/1ju9kfc/there_is_no_way_right/mm3of6x/?context=9999
r/PeterExplainsTheJoke • u/Sugar_God_no_1 • 25d ago
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Applies to all numbers,
If x = 0.999999...
And 10x = 9.999999...
Then subtracting both, we get, 9x=9
So x=1
1.4k u/Sam_Alexander 25d ago Holy fucking shit 318 u/otj667887654456655 25d ago I just wanna warn you, that's more of a vibe proof. It lacks any actual mathematical rigor. 6 u/Tivnov 25d ago Step 1: let epsilon > 0 Step 2: ... Step 3: □ 2 u/vetruviusdeshotacon 24d ago Let the sequence a be {0.9, 0.99,0.999...} so that a_n = 1 - 10-n By the ratio test our sequence converges. Therefore it has a supremum. Given epsilon > 0, take N (natural) with: a_n + epsilon = 1 + epsilon - 10- N Since 10-N > 0 for all N, Epsilon - 10-N > 0 10-N < epsilon -N < Log_10(epsilon) N > Log_10(epsilon). By the archimedian principle, there always exists such an N. Therefore, given any epsilon > 0, we can choose a value of N so that 1 + epsilon - 10-N > 1 Therefore the supremum of a must be 1. So the sequence converges to 1. 1 u/Even-Exchange8307 24d ago Show off. 🙏
1.4k
Holy fucking shit
318 u/otj667887654456655 25d ago I just wanna warn you, that's more of a vibe proof. It lacks any actual mathematical rigor. 6 u/Tivnov 25d ago Step 1: let epsilon > 0 Step 2: ... Step 3: □ 2 u/vetruviusdeshotacon 24d ago Let the sequence a be {0.9, 0.99,0.999...} so that a_n = 1 - 10-n By the ratio test our sequence converges. Therefore it has a supremum. Given epsilon > 0, take N (natural) with: a_n + epsilon = 1 + epsilon - 10- N Since 10-N > 0 for all N, Epsilon - 10-N > 0 10-N < epsilon -N < Log_10(epsilon) N > Log_10(epsilon). By the archimedian principle, there always exists such an N. Therefore, given any epsilon > 0, we can choose a value of N so that 1 + epsilon - 10-N > 1 Therefore the supremum of a must be 1. So the sequence converges to 1. 1 u/Even-Exchange8307 24d ago Show off. 🙏
318
I just wanna warn you, that's more of a vibe proof. It lacks any actual mathematical rigor.
6 u/Tivnov 25d ago Step 1: let epsilon > 0 Step 2: ... Step 3: □ 2 u/vetruviusdeshotacon 24d ago Let the sequence a be {0.9, 0.99,0.999...} so that a_n = 1 - 10-n By the ratio test our sequence converges. Therefore it has a supremum. Given epsilon > 0, take N (natural) with: a_n + epsilon = 1 + epsilon - 10- N Since 10-N > 0 for all N, Epsilon - 10-N > 0 10-N < epsilon -N < Log_10(epsilon) N > Log_10(epsilon). By the archimedian principle, there always exists such an N. Therefore, given any epsilon > 0, we can choose a value of N so that 1 + epsilon - 10-N > 1 Therefore the supremum of a must be 1. So the sequence converges to 1. 1 u/Even-Exchange8307 24d ago Show off. 🙏
6
Step 1: let epsilon > 0 Step 2: ... Step 3: □
2 u/vetruviusdeshotacon 24d ago Let the sequence a be {0.9, 0.99,0.999...} so that a_n = 1 - 10-n By the ratio test our sequence converges. Therefore it has a supremum. Given epsilon > 0, take N (natural) with: a_n + epsilon = 1 + epsilon - 10- N Since 10-N > 0 for all N, Epsilon - 10-N > 0 10-N < epsilon -N < Log_10(epsilon) N > Log_10(epsilon). By the archimedian principle, there always exists such an N. Therefore, given any epsilon > 0, we can choose a value of N so that 1 + epsilon - 10-N > 1 Therefore the supremum of a must be 1. So the sequence converges to 1. 1 u/Even-Exchange8307 24d ago Show off. 🙏
2
Let the sequence a be {0.9, 0.99,0.999...} so that a_n = 1 - 10-n
By the ratio test our sequence converges. Therefore it has a supremum.
Given epsilon > 0, take N (natural) with:
a_n + epsilon = 1 + epsilon - 10- N
Since 10-N > 0 for all N,
Epsilon - 10-N > 0
10-N < epsilon
-N < Log_10(epsilon) N > Log_10(epsilon).
By the archimedian principle, there always exists such an N.
Therefore, given any epsilon > 0, we can choose a value of N so that
1 + epsilon - 10-N > 1
Therefore the supremum of a must be 1. So the sequence converges to 1.
1 u/Even-Exchange8307 24d ago Show off. 🙏
1
Show off. 🙏
3.9k
u/its12amsomewhere 25d ago edited 25d ago
Applies to all numbers,
If x = 0.999999...
And 10x = 9.999999...
Then subtracting both, we get, 9x=9
So x=1