Why does this limit of this integral work?

"If we set epsilon > 0 and N as a large natural number, and if j, k > N:

OK where exactly is this coming from? This appears as one of the initial statements in a proof about f_j's integral approaching f's integral. But I could not find the theorem that states this. Is it supposed to be self-evident or something? :confused:

Re: Why does this limit of this integral work?

all you've written here is the triangle inequality where $\left|\displaystyle{\int_a^b}g(t)~dt\right|$ is a norm of $g(t)$

Re: Why does this limit of this integral work?

Quote:

Originally Posted by

**romsek** all you've written here is the triangle inequality where $\left|\displaystyle{\int_a^b}g(t)~dt\right|$ is a norm of $g(t)$

Ah, OK. The triangle inequality for integrals. Got it.

Re: Why does this limit of this integral work?

It's a little more complicated than that. Using the form of the triangle inequality ,

And using gives

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