This question has been asked multiple times, but none of them use the same method I have used to attempt the proof.
Here's what I did:
$s_n$ is a convergent sequence. Thus $$\left|s_n - s\right| \lt \epsilon$$ Reverse triangle inequality: $$\left|s_n\right| - \left|s\right| \le \left|s_n - s\right| \lt \epsilon$$ $$\text{or, } a \le\left|s_n\right| \lt \epsilon + \left|s\right|$$ $$\text{or, } a \lt \epsilon + |s| \text{ for arbitrary } \epsilon$$ $$\text{therefore, } a < \lim s_n$$ The problem here is that I am getting $a \lt \lim s_n$, whereas I should be getting $a \le \lim s_n$. Please tell me what I did wrong here.