TITLE: Show that $(x_n-y_n)$ converges to $x-y$.
QUESTION [5 upvotes]: Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$.

If it's asking about $(x_n+y_n)$. I know that I can use triangle inequality, but can I also use triangle inequality here? Thank you.

REPLY [2 votes]: Yes: 
$$|(x_n-y_n)-(x-y)|=|(x_n-x)-(y_n-y)|\le|x_n-x|+|y_n-y|\,.$$