Proposition 1.20

If $$f: X \rightarrow Y$$ is continuous and $$\lim_{n \rightarrow \infty}x_n = x$$ for $$x_1,x_2,\dots\in X$$ and $$x\in X$$, then $$\lim_{n\rightarrow\infty} f(x_n) = f(x)$$.

baruch_shahi
Let $$\epsilon > 0$$. Since $$f$$ is continuous, there exists $$\delta > 0$$ such that whenever $$d_X(x,y) < \delta$$ we have $$d_Y(f(x),f(y))< \epsilon.$$ Since $$x_n\to x$$ in $$X$$ there exists $$N$$ such that $$d_X(x_n,x)< \delta$$ for any $$n\ge N$$. By continuity, then, this immediately implies that $$d_Y(f(x_n),f(x))< \epsilon$$ for $$ n\geq$$ N. This exactly means that $$\lim\limits_{n\to\infty}{f(x_n)}= f(x)$$.

baruch_shahi 21:30, February 27, 2011 (UTC -5)

'''Exactly. I added "for $$ n\geq$$ N" for clarificaiton. --Steven.clontz 16:14, March 1, 2011 (UTC)'''