In his book Bauer proves the following theorem using the subsequence principle:
Proposition. If the sequence $(f_n)$ of measurable real functions on $\Omega$ converges stochastically to a measurable real function $f$ on $\Omega$, and $\phi:\mathbb{R}\to\mathbb{R}$ is continuous, then the sequence $(\phi\circ f_n)$ converges stochastically to $\phi\circ f$.
Here $(\Omega,\mathcal{A},\mu)$ is a fixed measure space and stochastic convergence means $\lim_{n\to\infty} \mu(\{|f_n-f|\geq \alpha\}\cap A)=0$ for all $\alpha>0$ and $A\in \mathcal{A}$ with $\mu(A)<\infty$.
As an exercise, Bauer asks to give an "elementary" proof of the previous propostion. He gives the following hint: Show that for each $\epsilon\in (0,1)$ there exist $\delta>0$ such that $$\{|f|\leq 1/\epsilon\}\cap \{|f_n-f|\leq \delta\}\subset \{|\phi\circ f_n-\phi\circ f| \leq \epsilon \}.$$
Using this I would have
$$\{|\phi\circ f_n-\phi\circ f| \geq \epsilon \} \subset \{|f|\geq 1/\epsilon\}\cup \{|f_n-f|\geq \delta\}$$
and so for $A\in \mathcal{A}$ of finite measure
$$\mu (\{|\phi\circ f_n-\phi\circ f| \geq \epsilon \}\cap A)\leq \mu( \{|f|\geq 1/\epsilon\}\cap A)+\mu(\{|f_n-f|\geq \delta\}\cap A).$$
The second term on the right tends to $0$ because of stochastic convergence, but what about the first term? Am I missing something?
Thanks a lot for your help.
Fix $k\in \mathbb{N}$. As $[-(k+1),(k+1)]$ is compact we have that $\phi$ is uniformly continuous on this interval. Hence given $\epsilon>0$ we can choose $0<\delta\leq 1$ such that $$ \{|f|\leq k\}\cap \{|f_n-f|< \delta\}\subset \{|\phi\circ f_n-\phi\circ f| <\epsilon \}$$ for all $n\in \mathbb{N}$. Hence for any $A\in\mathcal{A}$ of finite measure we have
$$\mu (\{|\phi\circ f_n-\phi\circ f| \geq \epsilon \}\cap A)\leq \mu( \{|f|> k\}\cap A)+\mu(\{|f_n-f|\geq \delta\}\cap A).$$
Letting $n\to\infty$ on both sides we get
$$\limsup_{n\to\infty}\mu (\{|\phi\circ f_n-\phi\circ f| \geq \epsilon \}\cap A)\leq \mu( \{|f|> k\}\cap A).$$
Now $\{|f|> k\}\cap A$ is a sequence of sets of finite measure decreasing to $\emptyset$, so letting $k\to\infty$ we finally obtain
$$\limsup_{n\to\infty}\mu (\{|\phi\circ f_n-\phi\circ f| \geq \epsilon \}\cap A)\leq 0.$$