1. absolutely convergent series

Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.

2. Originally Posted by Milus
Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.
Consider the Cauchy-Schwarz inequality.

3. Originally Posted by Mathstud28
Consider the Cauchy-Schwarz inequality.

Can you help me more.I really struggle with these proofs.The concept seems clear but I have no idea how to show it.

4. Originally Posted by Milus
Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.
Hint: For all sufficiently large n, $|a_n|$ must be less than 1, and therefore $a_n^2<|a_n|$.

5. Originally Posted by Opalg
Hint: For all sufficiently large n, $|a_n|$ must be less than 1, and therefore $a_n^2<|a_n|$.
Can you help me more.I just cannot connect it to one piece.

6. Originally Posted by Milus
Can you help me more.I just cannot connect it to one piece.
This is true: $\left| x \right| \leqslant 1 \Rightarrow \quad x^2 \leqslant x$.
Because $\left( {a_n } \right) \to 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {a_n } \right| < 1 \Rightarrow \quad \left( {a_n } \right)^2 \leqslant \left| {a_n } \right|} \right]$.

Now by direct comparison $\sum\limits_n {\left( {a_n } \right)^2 }$ must converge.

7. Is that sufficient proof or shoule there be added something????

8. Originally Posted by Milus
Is that sufficient proof or shoule there be added something????
Since for $n\geq N$ (sufficiently large $n$) we have that $a_n^2 \leq |a_n|$ it follows that $\sum_{n=N}^{\infty} a_n^2 \leq \sum_{n=N}^{\infty} |a_n| < \infty$ since $\{ a_n\}$ is absolutely convergent.