Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent
If $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent
I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous.     Statement: Let $(X_n)_{n \geq 1}$ be a sequence of non-negative random variables that are identically distributed $X_n \sim X \geq 0 $ with $E(X)&lt; \infty$, show that $ X_n/n \to 0$ almost surely     Proof: Since $X_n \sim X$ for some non negative random variables we can easily compute that for $\epsilon &gt;0$ we have $$ \infty&gt;E(X/\epsilon) = \int_0^\infty P(X/ \epsilon &gt; t) dt = \sum_{n=1}^\infty \int_{n-1}^n P(X/ \epsilon &gt; t ) dt \geq \sum_{n=1}^{\infty} P(X/ \epsilon &gt;n) $$ Thanks to the Borel-Cantelli Lemma we obtain that $$ P ( \limsup_{n \to \infty} \lbrace X_n/n &gt; \epsilon \rbrace ) = 0 \iff P ( \liminf_{n \to \infty} \lbrace X_n/n \leq \epsilon \rbrace) = 1 $$     Question: Does the $\epsilon$ in the above depend on $k$?   I think it does and I would argue as follows $$P( \liminf_{n \to \infty} \lbrace X_n/n \leq \epsilon \rbrace )= P  \left( \bigcup_{n=0}^\infty \bigcap_{k=n}  \lbrace X_k/k \leq \epsilon (k) \rbrace \right) $$ Now using the chain of inclusions that state that we have for $\epsilon &gt;0$  $$ \lbrace \limsup |X_n-X| &lt; \epsilon \rbrace \subset \liminf \lbrace |X_n-X| &lt; \epsilon \rbrace \subset \lbrace \limsup |X_n-X| \leq \epsilon \rbrace $$ We conclude that $$ P ( \lbrace \limsup_{n \to \infty } X_n/ n \leq \epsilon(k) \rbrace)=1 $$ Where I choose the same $k \in \mathbb{N}$ as above, and then choose $k^*$ large enough such that $\epsilon(k) \leq \frac{1}{k^*}$, define the set $$ \Lambda_{k^*}:= \lbrace \limsup_{n \to \infty} X_n/n \leq 1/k^* \rbrace \implies P(\Lambda_{k^*})=1$$ and also we obtain that $\Lambda:=\bigcap_{j \geq k^*} \Lambda_{k^*}$ is still an almost sure event, i.e. $P(\Lambda)=1$ but for all realizations $\omega \in \Lambda$ we have $\limsup_{n \to \infty} X_n( \omega)/n \leq 0  \implies X_n( \omega)/n \to 0$ for all $\omega \in \Lambda$ that is$X_n/n \to 0$ almost surely $\Box$  Are my arguments given above correct and more importantly complete? Are there steps where you'd use different arguments? tables with lots of columns I have a table that lists various attributes of an individuals personal development plan; objective, time to complete objective, comments etc. The table is editable so the user can directly add comments without having to go to an intermediate form. The problem is that the client wants to add more columns and this is messing up the visual display.   The alternatives I have are:   allowing columns to expand out on click keeping essential attributes in table, rest in 'advanced edit option form' or somesuch I don't know - have you any ideas?   One more thing to add: I think the addition of extra columns is based on user feedback ...  edit: here is what I have gone for mysql_fetch_array()/mysql_fetch_assoc()/mysql_fetch_row()/mysql_num_rows etc... expects parameter 1 to be resource So, my update to 13.04 didn't go very well. 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