Let f:D -> R be a continuous function. Let {x_{n}} be a sequence in D that converges to x in D. Then f(X_{n}) -> f(x)

[G-X]

Let [Math]f[/Math] : [Math]D[/Math] -> [Math]R[/Math] be a continuous function, where [Math]D \subset R^{k}[/Math]. Let [Math]\{x_{n}\}_{n = d}^{\infty}[/Math] be a sequence in [Math]D[/Math] that converges to [Math]x \in D[/Math]. Then [Math]f(X_{n})[/Math] -> [Math]f(x)[/Math]
We need to show that [Math]lim_{n -> \infty}f(x_{n}) = f(x)[/Math].

Since, [Math]\{x_{n}\}_{n = d}^{\infty}[/Math] is a sequence in [Math]D[/Math] that converges to [Math]x \in D[/Math] then there

[Math]\exists N_{\epsilon}[/Math] such that [Math]n \ge N_{\epsilon}[/Math] then [Math]|x - x_{n}| = r < \epsilon_{1}[/Math]. Since

[Math]\delta > 0, ((B_{\delta}(x) \cap D) / \{x\}) \neq \varnothing[/Math]. We know that there [Math]\exists \epsilon_{1} = \delta[/Math] meaning that

[Math]|x - x_{n}| = r < \delta[/Math], so [Math]\exists z = x_{n} \in B_{\delta}(x) \cap D[/Math].

Let [Math]a = x \in D[/Math], thus [Math]f[/Math] is continuous at [Math]x[/Math], by definition,

[Math]\forall z \in B_{\delta}(x) \cap D[/Math] s.t. [Math]|f(z) - f(x)| < \epsilon_{2}[/Math]. We know that [Math]\exists z = x_{n} \in B_{\delta}(x) \cap D[/Math],

so we can conclude that [Math]|f(x_{n}) - f(x)| < \epsilon{2}[/Math].

Thus we have proven that [Math]lim_{n -> \infty}f(x_{n}) = f(x)[/Math].

 

[G-X]

We need to show that [Math]lim_{n -> \infty}f(x_{n}) = f(x)[/Math].

By definition of convergence, [Math]|x - x_{n}| < \epsilon_{1}[/Math]. Since [Math]\delta > 0[/Math], this implies that [Math]((B_{\delta}(x) \cap D) / \{x\}) \neq \varnothing[/Math].

We know that by definition of convergence this statement holds [Math]\forall \epsilon_{1} > 0[/Math], so there [Math]\exists \epsilon_{1} = \delta[/Math] implying that [Math]|x - x_{n}| < \delta[/Math].

If the [Math]d(x, x_{n}) < \delta[/Math] then [Math]x_{n} \in B_{\delta}(x)[/Math]. We also know that [Math]x_{n} \in D[/Math], thus we can conclude that [Math]x_{n} \in B_{\delta}(x) \cap D[/Math].

Let [Math]a = x \in D[/Math], thus [Math]f[/Math] is continuous at [Math]x[/Math], by definition, [Math]\forall z \in B_{\delta}(x) \cap D[/Math] s.t. [Math]|f(z) - f(x)| < \epsilon_{2}[/Math].

We know that [Math]\exists z = x_{n} \in B_{\delta}(x) \cap D[/Math], so we can conclude that [Math]|f(x_{n}) - f(x)| < \epsilon{2}[/Math], [Math]\forall \epsilon_{2} > 0[/Math] by definition of continuous.

Thus we have proven that [Math]lim_{n -> \infty}f(x_{n}) = f(x)[/Math].

 

[Dr.Peterson]

I find this painful to read, because you are constantly referring to things that have not yet been defined. Before you mention epsilon, you should have said it can be any positive number. (That is, "Let [MATH]\epsilon_1 > 0[/MATH]".) Before you mention delta, you should define it. (In this case, the goal is to show that it exists, so you have to pick a value for it, not just let it be anything.)

And so on. In general, be much more careful about quantifiers. It's almost meaningless, for example, to say "[MATH]\exists\epsilon_1=\delta[/MATH]" (and the word "there" is redundant).

Once what you write becomes meaningful, we can check its meaning. Probably your flow of thought is right, but I can't yet say that.