Reference for "If $(a_n)$ is a sequence of real numbers with $\lim a_n^2=0$, then also $\lim a_n=0$"? Could you please give me a good reference as to the below-quoted real analysis result?


  If $(a_n)$ is a sequence of real numbers with $\lim a_n^2=0$, then also $\lim a_n=0$
By definition of a limit, for all $\varepsilon &gt; 0$ there exists $N \in \mathbb N$ such that $|a_n^2 - 0| &lt; \varepsilon$ for all $n &gt; N$.

Then we can see that the same holds for $a_n$: for any $\varepsilon &gt; 0$, we know that for $\varepsilon_1 = \varepsilon^2$ there exists $N \in \mathbb N$ such that $a_n^2 &lt; \varepsilon_1\hspace{0.2cm}\forall n&gt;N$, and this gives us $|a_n| &lt; \varepsilon\hspace{0.2cm}\forall n&gt;N$.
I am not sure, but if $\lim a_n=a,$ then if you divide $\lim a_n^2$ by $a,$ you get$$\frac{\lim a_n^2}{\lim a_n}=\lim\frac{a_n^2}{a_n}=\lim a_n=\frac{0}{a}=a.$$If you multiply both sides by $a,$ you get $a=0,$ but if $a=0,$ you can't multiply both sides by $a$. Since$$\lim_{\varepsilon\to0}\frac{\varepsilon}{a}=a\implies a^2=\lim_{\varepsilon\to0}\varepsilon,$$$a$ must converge to $0$.