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I'm not sure if I'm overcomplicating this, but I'm trying to prove that if $x_n \to x$ and $y_n \to y$, then $\lim_{n\to \infty} \rho(x_n, y_n) = \rho(x,y)$.

So far I have that I want to show that $\rho(\rho(x_n,y_n), \rho(x,y)) \to 0$, and I have tried a tricky triangle inequality: $$ \rho(\rho(x_n,y_n), \rho(x,y)) \leq \rho(\rho(x_n,y_n), \rho(x,y_n)) + \rho(\rho(x,y_n), \rho(x,y)) $$ But I'm pretty stuck here. A hint would be great. Thanks!

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Unless $\rho$ is a metric on $\Bbb R$, $\rho(\rho(x_n,y_n),\rho(x,y))$ doesn’t make sense; do you mean $|\rho(x_n,y_n)-\rho(x,y)|$? – Brian M. Scott Nov 5 '13 at 17:18
Ah yes I was unsure here whether I could use the absolute value metric $|\cdot|$ to prove convergence or whether I had to use a general $\rho$ to prove convergence. And $\rho$ is indeed a metric on $\mathbb R$ – Moderat Nov 5 '13 at 17:19
Is $\rho$ an arbitrary metric on $\Bbb R$, or some particular one? Either way, the convergence of $\rho(x_n,y_n)$ to $\rho(x,y)$ is convergence in the usual topology on $\Bbb R$, so you can use the usual $|\cdot - \cdot|$ metric. – Brian M. Scott Nov 5 '13 at 17:21
Hmm, in that case I suppose I'm just not quite sure how to show that something like $|\rho(x_n,y_n) - \rho(x, y_n)|$ can be made small? – Moderat Nov 5 '13 at 17:28
Hang on, and I’ll write up an answer. – Brian M. Scott Nov 5 '13 at 17:29
up vote 1 down vote accepted

$$\rho(x_n,y_n)\leq\rho(x_n,x)+\rho(x,y_n)\leq\rho(x_n,x)+\rho(x,y)+\rho(y,y_n)$$ $$\rho(x,y)\leq\rho(x,x_n)+\rho(x_n,y)\leq\rho(x,x_n)+\rho(x_n,y_n)+\rho(y_n,y)$$

Can you continue from here?

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