In Oksendal's book of Stochastic Differential Equations, the author does a reasoning similar to what's presented below.
The follwing three points are proved:
- I can find $\{\phi_n\}$ such that $E(\int_S^T(g-\phi_n)^2dt)\rightarrow 0$, as $n\rightarrow \infty$.
- I can find $\{g_n\}$ (each $g_n$ is a $g$ from point 1) such that $E(\int_S^T(h-g_n)^2dt)\rightarrow 0$, as $n\rightarrow \infty$.
- I can find $\{h_n\}$ (each $h_n$ is an $h$ from point 2) such that $E(\int_S^T(f-h_n)^2dt)\rightarrow 0$, as $n\rightarrow \infty$.
Then the author states that by the three points above, for any $f$ there is a sequence of $\{\phi_n\}$ such that $E(\int_S^T(f-\phi_n)^2dt)\rightarrow 0$, as $n\rightarrow \infty$.
How is he able to state that? I think he's probably using some simple relationship/inequality which allows him to conclude that, and I'm not getting it.
All of these functions have their own properties, stated in the book, which I didn't write, because in my opinion they would make the question bigger, without improving it.
He is essentially using the triangle inequality.
Fix some function $f$. Then by, 3., there is some $h_n$ with $\sqrt{\mathbb{E}\int_S^T (f-h_n)^2 \, dt} \leq \frac{1}{3n}$. Now, by 2., there is some $g_n$ with $\sqrt{\mathbb{E}\int_S^T (h_n-g_n)^2 \, dt} \leq \frac{1}{3n}$. Finally, by 1., there is $\phi_n$ with $\sqrt{\mathbb{E}\int_S^T (g_n-\phi_n)^2 \, dt} \leq \frac{1}{3n}$. Since $$\sqrt{\mathbb{E}\int_S^T f(t)^2 \, dt}$$ is a norm (the $L^2$-norm with respect to the product measure $\mathbb{P} \otimes \lambda$), it satisfies the triangle inequality. Writing $$f-\phi_n = (f-h_n)+(h_n-g_n)+(g_n-\phi_n)$$ we thus get
\begin{align*} \sqrt{\mathbb{E}\int_S^T (f-\phi_n)^2 \, dt} &\leq \sqrt{\mathbb{E}\int_S^T (f-h_n)^2 \, dt} + \sqrt{\mathbb{E}\int_S^T (h_n-g_n)^2 \, dt} \\ &\quad + \sqrt{\mathbb{E}\int_S^T (g_n-\phi_n)^2 \, dt} \\ &\leq 3 \frac{1}{3n}\end{align*}
and so $ \sqrt{\mathbb{E}\int_S^T (f-\phi_n)^2 \, dt} \to0$.
Remark: You can also read the statements a different (more analytic) way. Say, you have a normed space $X$ and two subsets $C$ and $D$. If $C$ is dense in $D$ and $D$ is dense in $X$, then $C$ is dense in $X$. Of course, you can iterate this (i.e. take another set which is dense in $C$ and so on). That's exactly Oksendal uses in his book. Statement 3. says that the functions $h$ (with certain properties) are dense (w.r.t. $L^2$-norm) in your original space of functions $f$. Statement 2. says that the functions $g$ (with certain properties) are dense (w.r.t. $L^2$-norm) in the set of functions $h$, and so on.