Question:
"Show that a progression $(a_n)_{n\ge1}$ applies for the following implication $$a_n \rightarrow a ~\text{when}~ n \rightarrow \infty, \Rightarrow |a_n| \rightarrow |a| ~\text{when}~ n \rightarrow \infty $$ Does the implication hold the other way?"
So I'm a bit stuck, first I don't really know how to show if the implication applies? Secondly, I've tried using the reverse triangle inequality when investigating if the implication holds the other way, which gives me
$$||a_n|-|a|| \le |a_n-a| $$ but then I'm also stuck as above. I assume that the implication doesn't hold, but I cannot seem to show it mathematically. Any advice?
Thanks, Sincerely.