Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.
This is true: $\displaystyle \left| x \right| \leqslant 1 \Rightarrow \quad x^2 \leqslant x$.
Because $\displaystyle \left( {a_n } \right) \to 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {a_n } \right| < 1 \Rightarrow \quad \left( {a_n } \right)^2 \leqslant \left| {a_n } \right|} \right]$.
Now by direct comparison $\displaystyle \sum\limits_n {\left( {a_n } \right)^2 } $ must converge.
Since for $\displaystyle n\geq N$ (sufficiently large $\displaystyle n$) we have that $\displaystyle a_n^2 \leq |a_n|$ it follows that $\displaystyle \sum_{n=N}^{\infty} a_n^2 \leq \sum_{n=N}^{\infty} |a_n| < \infty$ since $\displaystyle \{ a_n\}$ is absolutely convergent.