Pointwise limit function $f$ of sequence $(f_n)$
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My sequence of functions $$f_n (x) = begin{cases}
1 & ,x = frac{1}{n} \
x & ,x = 1,1/2, ...,1/(n-1) \
0 & ,otherwise end{cases}$$
My attempt is to fix $k in mathbb{N}$, consider the following cases when $x = 1/k$ for $n geq k $ and $x neq 1/k$. Is there a better approach to find the pointwise limit $f$ of this sequence $f_n$?
real-analysis
asked Nov 19 at 17:49
Dong Le
516
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up vote
1
down vote
favorite
My sequence of functions $$f_n (x) = begin{cases}
1 & ,x = frac{1}{n} \
x & ,x = 1,1/2, ...,1/(n-1) \
0 & ,otherwise end{cases}$$
My attempt is to fix $k in mathbb{N}$, consider the following cases when $x = 1/k$ for $n geq k $ and $x neq 1/k$. Is there a better approach to find the pointwise limit $f$ of this sequence $f_n$?
real-analysis
asked Nov 19 at 17:49
Dong Le
516
1
Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases.
– Math_QED
Nov 19 at 17:59
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
My sequence of functions $$f_n (x) = begin{cases}
1 & ,x = frac{1}{n} \
x & ,x = 1,1/2, ...,1/(n-1) \
0 & ,otherwise end{cases}$$
My attempt is to fix $k in mathbb{N}$, consider the following cases when $x = 1/k$ for $n geq k $ and $x neq 1/k$. Is there a better approach to find the pointwise limit $f$ of this sequence $f_n$?
real-analysis
asked Nov 19 at 17:49
Dong Le
516
My sequence of functions $$f_n (x) = begin{cases}
1 & ,x = frac{1}{n} \
x & ,x = 1,1/2, ...,1/(n-1) \
0 & ,otherwise end{cases}$$
My attempt is to fix $k in mathbb{N}$, consider the following cases when $x = 1/k$ for $n geq k $ and $x neq 1/k$. Is there a better approach to find the pointwise limit $f$ of this sequence $f_n$?
real-analysis
real-analysis
asked Nov 19 at 17:49
Dong Le
516
asked Nov 19 at 17:49
Dong Le
516
asked Nov 19 at 17:49
Dong Le
516
asked Nov 19 at 17:49
Dong Le
516
asked Nov 19 at 17:49
Dong Le
516
516
1
Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases.
– Math_QED
Nov 19 at 17:59
add a comment |
1
Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases.
– Math_QED
Nov 19 at 17:59
1
1
Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases.
– Math_QED
Nov 19 at 17:59
Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases.
– Math_QED
Nov 19 at 17:59
add a comment |
1 Answer
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1
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Let $x$ be a real.
If $x=frac 1k$ then for large enough $nge k+2,$ we will have
$$frac 1n <frac{1}{n-1}<xle 1$$ then
$$f_n(x)=x$$
and if $xne frac 1k implies f_n(x)=0$
The pointwise limit function is
$$f:xmapsto x$$
if $x=frac 1k$ and zero elsewhere.
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Let $x$ be a real.
If $x=frac 1k$ then for large enough $nge k+2,$ we will have
$$frac 1n <frac{1}{n-1}<xle 1$$ then
$$f_n(x)=x$$
and if $xne frac 1k implies f_n(x)=0$
The pointwise limit function is
$$f:xmapsto x$$
if $x=frac 1k$ and zero elsewhere.
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
add a comment |
up vote
1
down vote
Let $x$ be a real.
If $x=frac 1k$ then for large enough $nge k+2,$ we will have
$$frac 1n <frac{1}{n-1}<xle 1$$ then
$$f_n(x)=x$$
and if $xne frac 1k implies f_n(x)=0$
The pointwise limit function is
$$f:xmapsto x$$
if $x=frac 1k$ and zero elsewhere.
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
add a comment |
up vote
1
down vote
up vote
1
down vote
Let $x$ be a real.
If $x=frac 1k$ then for large enough $nge k+2,$ we will have
$$frac 1n <frac{1}{n-1}<xle 1$$ then
$$f_n(x)=x$$
and if $xne frac 1k implies f_n(x)=0$
The pointwise limit function is
$$f:xmapsto x$$
if $x=frac 1k$ and zero elsewhere.
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
Let $x$ be a real.
If $x=frac 1k$ then for large enough $nge k+2,$ we will have
$$frac 1n <frac{1}{n-1}<xle 1$$ then
$$f_n(x)=x$$
and if $xne frac 1k implies f_n(x)=0$
The pointwise limit function is
$$f:xmapsto x$$
if $x=frac 1k$ and zero elsewhere.
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
answered Nov 19 at 18:04
hamam_Abdallah
37k21534
37k21534
add a comment |
add a comment |
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Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases.
– Math_QED
Nov 19 at 17:59