Find a counterexample that the sequence ${f(a_n)}_n$ is bounded but not convergent
up vote
1
down vote
favorite
If $f :mathbb{R} rightarrow mathbb{R}$ be a strictly increasing continuous function and ${a_n}$ is a sequence in $[0,1]$ , then find a counterexample that the sequence ${f(a_n)}_n$ is bounded but not convergent?
I take $f(x) = e^x$ but this is not bounded . Again I take $f(x)= e^{-x}$ but this is not strictly increasing.
I'm not able to find a counterexample which satisfied the given statement.
Any hints/solution. Thanks.
real-analysis
edited Nov 17 at 11:35
Robert Z
91.1k1058129
asked Nov 17 at 11:28
Messi fifa
50111
add a comment |
up vote
1
down vote
favorite
If $f :mathbb{R} rightarrow mathbb{R}$ be a strictly increasing continuous function and ${a_n}$ is a sequence in $[0,1]$ , then find a counterexample that the sequence ${f(a_n)}_n$ is bounded but not convergent?
I take $f(x) = e^x$ but this is not bounded . Again I take $f(x)= e^{-x}$ but this is not strictly increasing.
I'm not able to find a counterexample which satisfied the given statement.
Any hints/solution. Thanks.
real-analysis
edited Nov 17 at 11:35
Robert Z
91.1k1058129
asked Nov 17 at 11:28
Messi fifa
50111
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $f :mathbb{R} rightarrow mathbb{R}$ be a strictly increasing continuous function and ${a_n}$ is a sequence in $[0,1]$ , then find a counterexample that the sequence ${f(a_n)}_n$ is bounded but not convergent?
I take $f(x) = e^x$ but this is not bounded . Again I take $f(x)= e^{-x}$ but this is not strictly increasing.
I'm not able to find a counterexample which satisfied the given statement.
Any hints/solution. Thanks.
real-analysis
edited Nov 17 at 11:35
Robert Z
91.1k1058129
asked Nov 17 at 11:28
Messi fifa
50111
If $f :mathbb{R} rightarrow mathbb{R}$ be a strictly increasing continuous function and ${a_n}$ is a sequence in $[0,1]$ , then find a counterexample that the sequence ${f(a_n)}_n$ is bounded but not convergent?
I take $f(x) = e^x$ but this is not bounded . Again I take $f(x)= e^{-x}$ but this is not strictly increasing.
I'm not able to find a counterexample which satisfied the given statement.
Any hints/solution. Thanks.
real-analysis
real-analysis
edited Nov 17 at 11:35
Robert Z
91.1k1058129
asked Nov 17 at 11:28
Messi fifa
50111
edited Nov 17 at 11:35
Robert Z
91.1k1058129
asked Nov 17 at 11:28
Messi fifa
50111
edited Nov 17 at 11:35
Robert Z
91.1k1058129
edited Nov 17 at 11:35
Robert Z
91.1k1058129
edited Nov 17 at 11:35
Robert Z
91.1k1058129
91.1k1058129
asked Nov 17 at 11:28
Messi fifa
50111
asked Nov 17 at 11:28
Messi fifa
50111
asked Nov 17 at 11:28
Messi fifa
50111
50111
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
3
down vote
accepted
Take $a_n:=frac{1+(-1)^n}{2}$ and any strictly increasing function $f$ in $[0,1]$.
answered Nov 17 at 11:33
Robert Z
91.1k1058129
1
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
add a comment |
up vote
3
down vote
Define $a_n=1$ if $n$ is even and $a_n=0$ if $n$ is odd.
answered Nov 17 at 11:33
drhab
94.9k543125
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Take $a_n:=frac{1+(-1)^n}{2}$ and any strictly increasing function $f$ in $[0,1]$.
answered Nov 17 at 11:33
Robert Z
91.1k1058129
1
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
add a comment |
up vote
3
down vote
accepted
Take $a_n:=frac{1+(-1)^n}{2}$ and any strictly increasing function $f$ in $[0,1]$.
answered Nov 17 at 11:33
Robert Z
91.1k1058129
1
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Take $a_n:=frac{1+(-1)^n}{2}$ and any strictly increasing function $f$ in $[0,1]$.
answered Nov 17 at 11:33
Robert Z
91.1k1058129
Take $a_n:=frac{1+(-1)^n}{2}$ and any strictly increasing function $f$ in $[0,1]$.
answered Nov 17 at 11:33
Robert Z
91.1k1058129
answered Nov 17 at 11:33
Robert Z
91.1k1058129
answered Nov 17 at 11:33
Robert Z
91.1k1058129
answered Nov 17 at 11:33
Robert Z
91.1k1058129
91.1k1058129
1
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
add a comment |
1
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
1
1
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
... or in fact any function $f$ with $f(0)ne f(1)$
– Hagen von Eitzen
Nov 17 at 11:38
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
@HagenvonEitzen I agree. Thanks for pointing out.
– Robert Z
Nov 17 at 11:40
add a comment |
up vote
3
down vote
Define $a_n=1$ if $n$ is even and $a_n=0$ if $n$ is odd.
answered Nov 17 at 11:33
drhab
94.9k543125
add a comment |
up vote
3
down vote
Define $a_n=1$ if $n$ is even and $a_n=0$ if $n$ is odd.
answered Nov 17 at 11:33
drhab
94.9k543125
add a comment |
up vote
3
down vote
up vote
3
down vote
Define $a_n=1$ if $n$ is even and $a_n=0$ if $n$ is odd.
answered Nov 17 at 11:33
drhab
94.9k543125
Define $a_n=1$ if $n$ is even and $a_n=0$ if $n$ is odd.
answered Nov 17 at 11:33
drhab
94.9k543125
answered Nov 17 at 11:33
drhab
94.9k543125
answered Nov 17 at 11:33
drhab
94.9k543125
answered Nov 17 at 11:33
drhab
94.9k543125
94.9k543125
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002240%2ffind-a-counterexample-that-the-sequence-fa-n-n-is-bounded-but-not-conver%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
