A description/proof of that limit condition can be substituted with with-equal conditions
Topics
About: limit
The table of contents of this article
Starting Context
- The reader knows a definition of limit.
Target Context
- The reader will have a description and a proof of the proposition that limit \(\delta - \varepsilon\) condition can be substituted with with-equal conditions.
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.
Main Body
1: Description
For any normed spaces map limit, \(lim_{v \Rightarrow v_0} f (v) = l\), the usual \(\delta - \varepsilon\) condition, for any \(0 \lt \varepsilon\), there is a \(0 \lt \delta\) such that for \(\{\forall v| \Vert v - v_0\Vert \lt \delta\}\), \(\Vert f (v) - l\Vert \lt \varepsilon\), can be substituted with the condition (called the 2nd condition here), for any \(0 \lt \varepsilon\), there is a \(0 \lt \delta\) such that for \(\{\forall v| \Vert v - v_0\Vert \le \delta\}\), \(\Vert f (v) - l\Vert \lt \varepsilon\). Additionally, the last part can be substituted with \(\Vert f (v) - l\Vert \le \varepsilon\), called the 3rd condition here.
2: Proof
Supposing the usual condition, there is a \(0 \lt \delta '\) such that for \(\{\forall v| \Vert v - v_0\Vert \lt \delta '\}\), \(\Vert f (v) - l\Vert \lt \varepsilon\), but it may be that \(\Vert f (v) - l\Vert \ge \varepsilon\) for a v such that \(\Vert v - v_0\Vert = \delta '\). Then, there is a \(0 \lt \delta = \delta ' - \lambda \lt \delta '\) where \(0 \lt \lambda\), then for \(\{\forall v| \Vert v - v_0\Vert \le \delta\}\), \(\Vert v - v_0\Vert \lt \delta '\), so, \(\Vert f (v) - l\Vert \lt \varepsilon\). On the other hand, supposing the 2nd condition, there is a \(0 \lt \delta '\) such that for \(\{\forall v| \Vert v - v_0\Vert \le \delta '\}\), \(\Vert f (v) - l\Vert \lt \varepsilon\), and there is the \(\delta = \delta '\) such that for \(\{\forall v| \Vert v - v_0\Vert \lt \delta\}\), \(\Vert v - v_0\Vert \le \delta '\), so, \(\Vert f (v) - l\Vert \lt \varepsilon\). As for the 3rd condition, supposing the 2nd condition, the \(\delta\) for the 2nd condition satisfies \(\Vert f (v) - l\Vert \le \varepsilon\); supposing the 3rd condition, any \(\delta\) that satisfies \(\Vert f (v) - l\Vert \le \varepsilon - \lambda\) satisfies \(\Vert f (v) - l\Vert \lt \varepsilon\).