Mixing
Proving that $lim_{n to infty}t_n = s$
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I'm stuck on the following problem:
$s_n$ and $t_n$ are sequences, such that $s_n=t_n$ except for finitely many values of $n$. Explain why if $lim_{n rightarrow infty} s_n = s$, then also $lim_{n rightarrow infty}t_n = s$, using the definition of limit.
Actually, my problem is that I understand that thing, but I can't come up with the idea of using the definition (with $epsilon$ and $N$) to prove it.
Thank you in advance for any help!
real-analysis
edited Oct 26 at 15:51
Bungo
13.5k22147
asked Oct 26 at 15:18
Sergey Malinov
51
add a comment |
up vote
0
down vote
favorite
I'm stuck on the following problem:
$s_n$ and $t_n$ are sequences, such that $s_n=t_n$ except for finitely many values of $n$. Explain why if $lim_{n rightarrow infty} s_n = s$, then also $lim_{n rightarrow infty}t_n = s$, using the definition of limit.
Actually, my problem is that I understand that thing, but I can't come up with the idea of using the definition (with $epsilon$ and $N$) to prove it.
Thank you in advance for any help!
real-analysis
edited Oct 26 at 15:51
Bungo
13.5k22147
asked Oct 26 at 15:18
Sergey Malinov
51
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm stuck on the following problem:
$s_n$ and $t_n$ are sequences, such that $s_n=t_n$ except for finitely many values of $n$. Explain why if $lim_{n rightarrow infty} s_n = s$, then also $lim_{n rightarrow infty}t_n = s$, using the definition of limit.
Actually, my problem is that I understand that thing, but I can't come up with the idea of using the definition (with $epsilon$ and $N$) to prove it.
Thank you in advance for any help!
real-analysis
edited Oct 26 at 15:51
Bungo
13.5k22147
asked Oct 26 at 15:18
Sergey Malinov
51
I'm stuck on the following problem:
$s_n$ and $t_n$ are sequences, such that $s_n=t_n$ except for finitely many values of $n$. Explain why if $lim_{n rightarrow infty} s_n = s$, then also $lim_{n rightarrow infty}t_n = s$, using the definition of limit.
Actually, my problem is that I understand that thing, but I can't come up with the idea of using the definition (with $epsilon$ and $N$) to prove it.
Thank you in advance for any help!
real-analysis
real-analysis
edited Oct 26 at 15:51
Bungo
13.5k22147
asked Oct 26 at 15:18
Sergey Malinov
51
edited Oct 26 at 15:51
Bungo
13.5k22147
asked Oct 26 at 15:18
Sergey Malinov
51
edited Oct 26 at 15:51
Bungo
13.5k22147
edited Oct 26 at 15:51
Bungo
13.5k22147
edited Oct 26 at 15:51
Bungo
13.5k22147
13.5k22147
asked Oct 26 at 15:18
Sergey Malinov
51
asked Oct 26 at 15:18
Sergey Malinov
51
asked Oct 26 at 15:18
Sergey Malinov
51
51
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
0
down vote
accepted
Since you know that $lim _{nto infty } s_n =s$, given a positive $epsilon$ you can find an $N$ such that $$nge N implies |s_n - s | < epsilon$$
On the other hand you can find an $M$ such that $$ n>M implies s_n=t_n$$
Let $K= max {N,M }$ and see what happens if $n>K$
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
add a comment |
up vote
1
down vote
HINT
Since $s_n=t_n$ except for finitely many values of $n$ a maximum value $bar n$ exixts such that $s_{bar n}neq t_{bar n}$.
Then refer to the definition of limit.
answered Oct 26 at 15:39
gimusi
85.5k74294
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Since you know that $lim _{nto infty } s_n =s$, given a positive $epsilon$ you can find an $N$ such that $$nge N implies |s_n - s | < epsilon$$
On the other hand you can find an $M$ such that $$ n>M implies s_n=t_n$$
Let $K= max {N,M }$ and see what happens if $n>K$
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
add a comment |
up vote
0
down vote
accepted
Since you know that $lim _{nto infty } s_n =s$, given a positive $epsilon$ you can find an $N$ such that $$nge N implies |s_n - s | < epsilon$$
On the other hand you can find an $M$ such that $$ n>M implies s_n=t_n$$
Let $K= max {N,M }$ and see what happens if $n>K$
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Since you know that $lim _{nto infty } s_n =s$, given a positive $epsilon$ you can find an $N$ such that $$nge N implies |s_n - s | < epsilon$$
On the other hand you can find an $M$ such that $$ n>M implies s_n=t_n$$
Let $K= max {N,M }$ and see what happens if $n>K$
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
Since you know that $lim _{nto infty } s_n =s$, given a positive $epsilon$ you can find an $N$ such that $$nge N implies |s_n - s | < epsilon$$
On the other hand you can find an $M$ such that $$ n>M implies s_n=t_n$$
Let $K= max {N,M }$ and see what happens if $n>K$
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
edited 23 hours ago
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
answered Oct 26 at 15:47
Mohammad Riazi-Kermani
40.2k41958
40.2k41958
add a comment |
add a comment |
up vote
1
down vote
HINT
Since $s_n=t_n$ except for finitely many values of $n$ a maximum value $bar n$ exixts such that $s_{bar n}neq t_{bar n}$.
Then refer to the definition of limit.
answered Oct 26 at 15:39
gimusi
85.5k74294
add a comment |
up vote
1
down vote
HINT
Since $s_n=t_n$ except for finitely many values of $n$ a maximum value $bar n$ exixts such that $s_{bar n}neq t_{bar n}$.
Then refer to the definition of limit.
answered Oct 26 at 15:39
gimusi
85.5k74294
add a comment |
up vote
1
down vote
up vote
1
down vote
HINT
Since $s_n=t_n$ except for finitely many values of $n$ a maximum value $bar n$ exixts such that $s_{bar n}neq t_{bar n}$.
Then refer to the definition of limit.
answered Oct 26 at 15:39
gimusi
85.5k74294
HINT
Since $s_n=t_n$ except for finitely many values of $n$ a maximum value $bar n$ exixts such that $s_{bar n}neq t_{bar n}$.
Then refer to the definition of limit.
answered Oct 26 at 15:39
gimusi
85.5k74294
answered Oct 26 at 15:39
gimusi
85.5k74294
answered Oct 26 at 15:39
gimusi
85.5k74294
answered Oct 26 at 15:39
gimusi
85.5k74294
85.5k74294
add a comment |
add a comment |
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