Say I have a set of functions $f, f_{n} \in C([a,b])$, $f_{n} \leq f_{n+1}$ and $f_{n}(x_0) \rightarrow f(x_0)$ (pointwise convergence). Also define $g_{n}(x) = f(x) - f_{n}(x)$.
My objective is to show that the function $g_{n}$ is bounded by $\epsilon$. Mechanically I understand the process of what needs to be done. In particular:
i) since $g_{n}$ is continuous then there exists as $\delta > 0$ s.t if $|x_0 - y| \leq \delta \Rightarrow |g_{n}(x_{0}) - g(y)| < \frac{\epsilon}{2}$
ii) since $f_{n}(x_{0}) \rightarrow f(x_0)$ point wise, there is a $N > 0$ s.t. $|f_{n}(x_{0}) - f(x_{0})| < \frac{\epsilon}{2}$.
My question is what is being said when we use the triangle inequality? specifically (lot's of steps left out mechanics not important here):
$$|g_{n}(y)| = |f(y) - f_{n}(y)| \leq |f(x_0) - f_{n}(x_{0})| + |g_{n}(x_{0}) - g_{n}(y)| \leq \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$$
So would this translate to something of the form: "Since we have that our function converges point wise AND our function is continuous then our function is bounded above by $\epsilon$." ?
Like I said I understand the mechanics of going through how to solve problems of this sort, but I'm trying to grasp the deeper notions. Because what I see are two conditions we established and somehow putting them together to arrive at a result.
If there is any more clarification needed in what I'm trying to ask just leave a comment.
In a sense, yes. The conclusion would be, since the function $f_n$ converges pointwise and is continuous on $[a, b]$, the difference between $f_{n}(y)$ and $f(y)$, where $y \in [a,b]$, is bounded by $\epsilon$. Hence, $g_{n}(y)$ is bounded by $\epsilon$.