00:01
So the idea behind this one is to use the finite version of it.
00:05
So first, sorry, uh, so first let's take from i equals, i guess, zero, or one.
00:22
I equals one or k, let's use case to see the same notation to sum m, right, of x k.
00:33
And now you know that this is a, uh, this is a final.
00:37
It some of things so this is like x1 plus x2 plus all the way to plus xm and since we have a norm you can use triangle inequality so by triangle inequality this is for example you can split this up if you want to be more pedantic you can split this up as x1 plus all of these terms this a vector space so this will be x1 norm of x1 plus norm of x2 plus all the way to xn and then you can keep repeating this so you would do the same thing with x2 to x10 so in the end what you get is that this is less than by triangle inequality this is less than equal to k equals 1 to n of the norm of x k and since you're always since now what about any end from to infinity well note that you can always add them right and make it larger so to this you can always you can always make it larger by making another less than equaled sign by adding the rest of the terms from n to infinity so this is always bigger than or equal to k equals one to infinity of the norm of x k right and this is almost what you want to show and how do you show this, well, this is almost what you want to show.
02:22
You just need to have infinity on the left -hand side too.
02:26
But so the argument for that is to say that, well, this holds this inequality.
02:31
So what have we shown? we've shown that take any n, so for every n and n, we've shown that the sum, the norm of the sum, k equals 1 to n, is less than equal to sum of k equals 1 to infinity of the norm of xk.
02:54
So we've shown this...