Why we can always choose $n_{k+1}>n_k$?
Multi tool use
$begingroup$
Let $E$ is a complex inner product space and $(x_n)_{n},(y_n)_{n}subseteq E$ are unit sequences.
If we suppose that it is not true that there is a constant $c<1$ such that
$$|langle x_n|y_nrangle|leq c <1 ,$$
for all $n$ sufficiently large.
Therefore, for every positive integer $k$, the number $c_k=1-1/k$ does not satisfy
$$|langle x_n|y_nrangle|leq 1-1/k ,tag{*}$$
for all $n$ sufficiently large, which means that it is not true that there exists an $N$ such that for all $n>N$ the inequality $(*)$ is valid.
Therefore, there exists some $n_k$ such that
$$1geq |langle x_{n_k}|y_{n_k}rangle|> 1-1/k, $$
where the inequality $1geq |langle x_{n_k}|y_{n_k}rangle|$ follows from the fact that $x_n,y_n$ are unit vectors.
Hence
$$lim_{ktoinfty}|langle x_{n_k}|y_{n_k}rangle|=1.$$
Why we can always choose $n_{k+1}>n_k$?
real-analysis
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
$endgroup$
add a comment |
$begingroup$
Let $E$ is a complex inner product space and $(x_n)_{n},(y_n)_{n}subseteq E$ are unit sequences.
If we suppose that it is not true that there is a constant $c<1$ such that
$$|langle x_n|y_nrangle|leq c <1 ,$$
for all $n$ sufficiently large.
Therefore, for every positive integer $k$, the number $c_k=1-1/k$ does not satisfy
$$|langle x_n|y_nrangle|leq 1-1/k ,tag{*}$$
for all $n$ sufficiently large, which means that it is not true that there exists an $N$ such that for all $n>N$ the inequality $(*)$ is valid.
Therefore, there exists some $n_k$ such that
$$1geq |langle x_{n_k}|y_{n_k}rangle|> 1-1/k, $$
where the inequality $1geq |langle x_{n_k}|y_{n_k}rangle|$ follows from the fact that $x_n,y_n$ are unit vectors.
Hence
$$lim_{ktoinfty}|langle x_{n_k}|y_{n_k}rangle|=1.$$
Why we can always choose $n_{k+1}>n_k$?
real-analysis
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
$endgroup$
add a comment |
$begingroup$
Let $E$ is a complex inner product space and $(x_n)_{n},(y_n)_{n}subseteq E$ are unit sequences.
If we suppose that it is not true that there is a constant $c<1$ such that
$$|langle x_n|y_nrangle|leq c <1 ,$$
for all $n$ sufficiently large.
Therefore, for every positive integer $k$, the number $c_k=1-1/k$ does not satisfy
$$|langle x_n|y_nrangle|leq 1-1/k ,tag{*}$$
for all $n$ sufficiently large, which means that it is not true that there exists an $N$ such that for all $n>N$ the inequality $(*)$ is valid.
Therefore, there exists some $n_k$ such that
$$1geq |langle x_{n_k}|y_{n_k}rangle|> 1-1/k, $$
where the inequality $1geq |langle x_{n_k}|y_{n_k}rangle|$ follows from the fact that $x_n,y_n$ are unit vectors.
Hence
$$lim_{ktoinfty}|langle x_{n_k}|y_{n_k}rangle|=1.$$
Why we can always choose $n_{k+1}>n_k$?
real-analysis
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
$endgroup$
Let $E$ is a complex inner product space and $(x_n)_{n},(y_n)_{n}subseteq E$ are unit sequences.
If we suppose that it is not true that there is a constant $c<1$ such that
$$|langle x_n|y_nrangle|leq c <1 ,$$
for all $n$ sufficiently large.
Therefore, for every positive integer $k$, the number $c_k=1-1/k$ does not satisfy
$$|langle x_n|y_nrangle|leq 1-1/k ,tag{*}$$
for all $n$ sufficiently large, which means that it is not true that there exists an $N$ such that for all $n>N$ the inequality $(*)$ is valid.
Therefore, there exists some $n_k$ such that
$$1geq |langle x_{n_k}|y_{n_k}rangle|> 1-1/k, $$
where the inequality $1geq |langle x_{n_k}|y_{n_k}rangle|$ follows from the fact that $x_n,y_n$ are unit vectors.
Hence
$$lim_{ktoinfty}|langle x_{n_k}|y_{n_k}rangle|=1.$$
Why we can always choose $n_{k+1}>n_k$?
real-analysis
real-analysis
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
asked Jan 9 at 15:25
SchülerSchüler
1,4481421
1,4481421
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The contrapositive of the statement "There exists $NinBbb N$ such that
$$
|langle x_n|y_nrangle|leq 1-1/k
$$
holds for all $nge N$" is
"For each $Nin Bbb N$, we can find $m > N$ such that
$$
|langle x_m|y_mrangle| > 1-1/k
$$
holds."
Now, we just take $N=n_k$ and take $n_{k+1}$ to be $m$ from the above statement.
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
$endgroup$
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3067571%2fwhy-we-can-always-choose-n-k1n-k%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The contrapositive of the statement "There exists $NinBbb N$ such that
$$
|langle x_n|y_nrangle|leq 1-1/k
$$
holds for all $nge N$" is
"For each $Nin Bbb N$, we can find $m > N$ such that
$$
|langle x_m|y_mrangle| > 1-1/k
$$
holds."
Now, we just take $N=n_k$ and take $n_{k+1}$ to be $m$ from the above statement.
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
$endgroup$
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
add a comment |
$begingroup$
The contrapositive of the statement "There exists $NinBbb N$ such that
$$
|langle x_n|y_nrangle|leq 1-1/k
$$
holds for all $nge N$" is
"For each $Nin Bbb N$, we can find $m > N$ such that
$$
|langle x_m|y_mrangle| > 1-1/k
$$
holds."
Now, we just take $N=n_k$ and take $n_{k+1}$ to be $m$ from the above statement.
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
$endgroup$
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
add a comment |
$begingroup$
The contrapositive of the statement "There exists $NinBbb N$ such that
$$
|langle x_n|y_nrangle|leq 1-1/k
$$
holds for all $nge N$" is
"For each $Nin Bbb N$, we can find $m > N$ such that
$$
|langle x_m|y_mrangle| > 1-1/k
$$
holds."
Now, we just take $N=n_k$ and take $n_{k+1}$ to be $m$ from the above statement.
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
$endgroup$
The contrapositive of the statement "There exists $NinBbb N$ such that
$$
|langle x_n|y_nrangle|leq 1-1/k
$$
holds for all $nge N$" is
"For each $Nin Bbb N$, we can find $m > N$ such that
$$
|langle x_m|y_mrangle| > 1-1/k
$$
holds."
Now, we just take $N=n_k$ and take $n_{k+1}$ to be $m$ from the above statement.
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
answered Jan 9 at 15:39
BigbearZzzBigbearZzz
8,49221652
8,49221652
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
add a comment |
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
Thank you for answer. Please how can I write correctely the sentence ''there is a constant $theta<1$ such that $|langle x_n|y_nrangle|leq c <1 $ for all $n$ sufficiently large.'' ? It is true to write ''there is a constant $theta<1$ and $Nin mathbb{N}$ such that for all $ngeq N$ we have $$|langle x_n|y_nrangle|leq c <1. $$ ?
$endgroup$
– Schüler
Jan 9 at 15:55
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
$begingroup$
@Schüler You are correct.
$endgroup$
– BigbearZzz
Jan 9 at 16:12
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.
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%2f3067571%2fwhy-we-can-always-choose-n-k1n-k%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
iaahXqye8MnXnZqz,EIm5
