Could someone please help me to clarify the logic of what is the WTS part for the reverse order of proof for the following IFF proposition please?
Proposition: Let X $\subset$ $\mathbf{R^n}$. A function f: X -> $\mathbf{R^m}$ is continuous at $x_0$ $\in$ X IFF for every sequence $x_i$ $\in$ X converging to $x_0$, $\lim_{i\to\infty}f(x_i)$ = $f(x_0)$
Thanks
Sorry i am not quite sure the logic to proof this specific IFF proposition of... need someone to help me sort the maths logic out: what would be the propositions that we know and what would be the part that we want to proof.
Before we begin, you should avoid using acronyms and abbreviations when posting questions, as it could be idiosyncratic to your course or text.
You have a proposition you wish to prove:
Let $X\subset \mathbb{R}^n$. A function $f:X\rightarrow \mathbb{R}^m$ is continuous at $x_0\in X$ if and only if for every sequence $\{x_i\}\in X^{\infty}$ which converges to $x_0$ we have that $\lim\limits_{i\rightarrow \infty}f(x_i)=f(x_0)$.
When dealing with a statement containing an "if and only if", you are secretly dealing with two statements. If we envision your original proposition as "$P$ if and only if $Q$" then what you will want to consider (usually) are the two related statements "if $P$ then $Q$" and "if $Q$ then $P$".
Here, that corresponds to the following:
1) Let $X\subset \mathbb{R}^n$. If a function $f:X\rightarrow \mathbb{R}^m$ is continuous at $x_0\in X$, then for every sequence $\{x_i\}\in X^{\infty}$ which converges to $x_0$ we have that $\lim\limits_{i\rightarrow \infty}f(x_i)=f(x_0)$.
2) Let $X\subset \mathbb{R}^n$. If for every sequence $\{x_i\}\in X^{\infty}$ which converges to $x_0$ we have that $\lim\limits_{i\rightarrow \infty}f(x_i)=f(x_0)$, then $f:X\rightarrow \mathbb{R}^m$ is continuous at $x_0\in X$.