Proof of convergent sequence

Question

I have the issue of how one would prove that if you have a convergent sequence $\{a_n\}$ with $\lim_{n\rightarrow\infty}a_n=a$, then the sequence ${−a_n}$ is also convergent and satisfies $\lim_{n\rightarrow\infty} −a_n=−a$? My attempt at this has been to go with $|−a_n−(−a)|<\varepsilon$, which becomes positive since it is absolute value, but i am not quite sure that this is right.

Answer 1

Yes that’s fine indeed we simply have that

$$|a_n-a|= |-(a_n-a)|= |-a_n-(-a)|<\varepsilon$$