49: Limit Condition of Normed Vectors Spaces Map Can Be Substituted with With-Equal Conditions

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description/proof of that limit condition of normed vectors spaces map can be substituted with with-equal conditions

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of limit of normed vectors spaces map at point.

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Target Context

• The reader will have a description and a proof of the proposition that the limit $\delta -ϵ$ condition of normed vectors spaces map can be substituted with with-equal conditions.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V_1$: $\in \{\text{ the normed vectors spaces }\}$
$V_2$: $\in \{\text{ the normed vectors spaces }\}$
$f$: $:V_1\to V_2$
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Statements:
$li{m}_{v\to v_0}f(v)=l$
$⟺$
$\mathrm{\forall }ϵ\in \mathbb{R}\text{ such that }0<ϵ(\mathrm{\exists }\delta \in \mathbb{R}\text{ such that }0<\delta (\mathrm{\forall }v\in V_1\text{ such that }\Vert v-v_0\Vert <\delta (\Vert f(v)-l\Vert <ϵ)))$ (the usual condition)
$⟺$
$\mathrm{\forall }ϵ\in \mathbb{R}\text{ such that }0<ϵ(\mathrm{\exists }\delta \in \mathbb{R}\text{ such that }0<\delta (\mathrm{\forall }v\in V_1\text{ such that }\Vert v-v_0\Vert \le \delta (\Vert f(v)-l\Vert <ϵ)))$ (the 2nd condition)
$⟺$
$\mathrm{\forall }ϵ\in \mathbb{R}\text{ such that }0<ϵ(\mathrm{\exists }\delta \in \mathbb{R}\text{ such that }0<\delta (\mathrm{\forall }v\in V_1\text{ such that }\Vert v-v_0\Vert <\delta (\Vert f(v)-l\Vert \le ϵ)))$ (the 3rd condition)
$⟺$
$\mathrm{\forall }ϵ\in \mathbb{R}\text{ such that }0<ϵ(\mathrm{\exists }\delta \in \mathbb{R}\text{ such that }0<\delta (\mathrm{\forall }v\in V_1\text{ such that }\Vert v-v_0\Vert \le \delta (\Vert f(v)-l\Vert \le ϵ)))$ (the 4th condition)
//

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2: Natural Language Description

For any normed vectors spaces, $V_1,V_2$, any map, $f:V_1\to V_2$, any limit, $li{m}_{v\to v_0}f(v)=l$, and the usual $\delta -ϵ$ condition, 'for each $0<ϵ$, there is a $0<\delta$ such that for each $v$ such that $\Vert v-v_0\Vert <\delta$, $\Vert f(v)-l\Vert <ϵ$', the "$\Vert v-v_0\Vert <\delta$" part can be substituted with $\Vert v-v_0\Vert \le \delta$ (called the 2nd condition here); or the "$\Vert f(v)-l\Vert <ϵ$" part can be substituted with $\Vert f(v)-l\Vert \le ϵ$ (called the 3rd condition here); or the both parts can be substituted (called the 4th condition here).

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3: Proof

Let us suppose the usual condition and see that the 2nd condition is satisfied.

Let $0<ϵ$ be any.

There is a $0<{\delta }^{\prime }$ such that for each $v$ such that $\Vert v-v_0\Vert <{\delta }^{\prime }$, $\Vert f(v)-l\Vert <ϵ$.

Let us choose any $0<\delta <{\delta }^{\prime }$.

Then, for each $v$ such that $\Vert v-v_0\Vert \le \delta$, $\Vert v-v_0\Vert \le \delta <{\delta }^{\prime }$, so, $\Vert f(v)-l\Vert <ϵ$.

So, the 2nd condition is satisfied.

Let us suppose the 2nd condition and see that the usual condition is satisfied.

Let $0<ϵ$ be any.

There is a $0<\delta$ such that for each $v$ such that $\Vert v-v_0\Vert \le \delta$, $\Vert f(v)-l\Vert <ϵ$.

Then, for each $v$ such that $\Vert v-v_0\Vert <\delta$, $\Vert v-v_0\Vert <\delta \le \delta$, so, $\Vert f(v)-l\Vert <ϵ$.

So, the usual condition is satisfied.

Let us suppose the usual condition and see that the 3rd condition is satisfied.

Let $0<ϵ$ be any.

There is a $0<\delta$ such that for each $v$ such that $\Vert v-v_0\Vert <\delta$, $\Vert f(v)-l\Vert <ϵ$.

Then, $\Vert f(v)-l\Vert <ϵ\le ϵ$.

So, the 3rd condition is satisfied.

Let us suppose the 3rd condition and see that the usual condition is satisfied.

Let $0<ϵ$ be any.

Let us choose any $0<{ϵ}^{\prime }<ϵ$.

There is a $0<\delta$ such that for each $v$ such that $\Vert v-v_0\Vert <\delta$, $\Vert f(v)-l\Vert \le {ϵ}^{\prime }$.

Then, $\Vert f(v)-l\Vert \le {ϵ}^{\prime }<ϵ$.

So, the usual condition is satisfied.

Let us suppose the 2nd condition and see that the 4th condition is satisfied.

Let $0<ϵ$ be any.

There is a $0<\delta$ such that for each $v$ such that $\Vert v-v_0\Vert \le \delta$, $\Vert f(v)-l\Vert <ϵ$.

Then, $\Vert f(v)-l\Vert <ϵ\le ϵ$.

So, the 4th condition is satisfied.

Let us suppose the 4th condition and see that the 2nd condition is satisfied.

Let $0<ϵ$ be any.

Let us choose any $0<{ϵ}^{\prime }<ϵ$.

There is a $0<\delta$ such that for each $v$ such that $\Vert v-v_0\Vert \le \delta$, $\Vert f(v)-l\Vert \le {ϵ}^{\prime }$.

Then, $\Vert f(v)-l\Vert \le {ϵ}^{\prime }<ϵ$.

So, the 2nd condition is satisfied.

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References

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