I was hoping I could get some feedback on my proof that a function that is Lipschitz is continuous. I am at the undergraduate level going through Carothers, chapter 5 if you are curious!
Suppose $f:(M,d)\rightarrow (N,\rho)$ is Lipschitz. Then for every $x,y\in M$ there is a $K<\infty$ such that $\rho (f(x),f(y))\leq Kd(x,y).$ Let $x_n\rightarrow x$ and $y_n\rightarrow y$ in M. Then we have that $\rho (f(x_n),f(y_n))\leq Kd(x_n,y_n)$. Therefore by the sequential characterization of continuity $f$ is continuous.
thanks again for your help!
Continuity does not require two sequences, but just one. Therefore, $x$ stays $x$, and only choose $y = x_n$, with $x_n \to x$. Then, indeed, $d(f(x), f(x_n)) \le K \rho (x, x_n)$, and since $x_n \to x$, you have $\rho (x, x_n) \to 0$, therefore $d(f(x), f(x_n)) \to 0$, which means $f(x_n) \to f(x)$, so indeed $f$ is continuous.
You would have needed two sequences if you had been asked to show that $f$ is uniformly continuous (and, indeed, this is what you prove in your post). In particular, since uniform continuity implies continuity, your proof is correct too - but it is more than what you were required to show.