Prove the following theorem Thm (Hw Problem - pf) Component Nature oe Convergence]Let PiAR be thejth Component fn of f.Let ē bealimit point of A.dim f (=)=L=(4,.., lm) iff tjc E, m}dim f;(x)-Jj%3D

Given function is f:A⊂ℝn→ℝm and fj:A⊂ℝn→ℝm be jth component. Let P be a limit point of A. Case…

Answered: Thm (Hw Problem - pf)Component Nature…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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Prove the following theorem

Thm (Hw Problem - pf) Component Nature oe Convergence]
Let PiA<R- BM Let f;AsR>R be the
jth Component fn of f.
Let ē bea
limit point of A.
dim f (=)=L=(4,.., lm) iff tjc E, m}
dim f;(x)-Jj
%3D

Expert Solution

Step 1

Given function is $f : A \subset ℝ^{n} \to ℝ^{m}$ and $f_{j} : A \subset ℝ^{n} \to ℝ^{m}$ be $j^{t h}$ component. Let $P$ be a limit point of $A$ .

Case 1(Forward direction proof):

Let $\lim_{\bar{x} \to P} f (\bar{x}) = L = (l_{1}, l_{2}, \dots, l_{m})$ . Therefore for each $ε > 0$ there exist a $δ > 0$ such that:

$||f (\bar{x}) - L|| < ε$ for all $\bar{x}$ satisfying $||\bar{x} - P|| < δ$ .

Consider the $j^{t h}$ component of given function $f_{j} (\bar{x})$ .

Since $||f_{j} (\bar{x}) - l_{j}|| \leq ||f (\bar{x}) - L||$ , therefore for each $ε > 0$ , there exist a $δ > 0$ such that:

$\begin{array}{rcl} ||f_{j} (\bar{x}) - l_{j}|| & \leq & ||f (\bar{x}) - L|| \\ < & ε \end{array}$

Whenever $||\bar{x} - P|| < δ$ . Hence $\lim_{\bar{x} \to P} f_{j} (\bar{x}) = l_{j}$ .

Since $j \in \{1, 2, \dots, m\}$ was arbitrary, therefore $\lim_{\bar{x} \to P} f_{j} (\bar{x}) = l_{j}$ for all $j \in \{1, 2, \dots, m\}$ .

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Thm (Hw Problem - pf)Component Nature Convergence] Let PiA C R"- BM Let f; AsR ,B be the Sth component fr of f. limit point of A. dim f (<)= L- (e,.-., lm) iff Hjc {l, m3 스m 우, (x)-서