norm convergence law question

i noticed in this problem

that the norm gives us a series which is diverging.

then they proove that its a couchy series so its converging

why?????(i thought that if the sum of te norm is dievrgant then thats it,its divergant finily)

what is the law behind this whole thing?

http://i40.tinypic.com/rif5.jpg

Re: norm convergence law question

We have $\displaystyle \left\|\sum_{i=1}^n f_i\right\|\le\sum_{i=1}^n\|f_i\|$, so even when the right-hand side diverges, the left-hand side may still converge. E.g., if $\displaystyle f_n=\frac{(-1)^n}{n}\in L^2([0,1])$, then $\displaystyle \|f_n\|=1/n$ and $\displaystyle \sum_{i=1}^\infty \|f_i\|$ diverges but $\displaystyle \sum_{i=1}^\infty f_i$ converges by the alternating series test. This is similar how absolute convergence implies convergence but not vice versa for numerical series.

The figure in your post (at least the formulas) do not finish the proof that $\displaystyle \|S_m-S_n\|$ is small, i.e., that $\displaystyle S_n$ is a Cauchy sequence.

Re: norm convergence law question

but its not smaller then diverging series

its equal to a diverging series.

so if its equal to a diverging series,then that it.

how can they change that by proving that its a couchy series.

(i know the law that if its a couchy series and its on L2 then its converging)

but here it looks like they cheat because at first they showed that its diverging ,but then they change that around.

this is the rest of the proof.

http://i41.tinypic.com/zjcdxe.jpg