The functions $f,g$ are continuous in $[a, b]$, and they satisfy the properties:
$$f(a) = g(b) \\ f(b) = g(a)$$
Show that $f$ and $g$ intersect in the interval $[a,b]$.
My attempt:
Let $h(x) = f(x) - g(x)$. Because of the previous definitions we have that without loss of generality that $h(a) \leq 0 \leq h(b)$. It follows that $h$ is continuous. By the definition of continuity there is a $c$ in the interval such that $h(c) = 0$. Therefore rearranging $f(c) = g(c)$ and the proof is complete.
Does this work or am I missing something?
As you said, let $h(x)=f(x)-g(x)$ then we get: $$h(a)=f(a)-g(a)=f(a)-f(b)\\ h(b)=f(b)-g(b)=f(b)-f(a) $$ Therefore $h(a)=-h(b)$.
Since $h$ is continous, we know $$\exists c\in[a,b]\ \ s.t \ \ h(c)=0\implies f(c)=g(c)$$
Almost identical to your proof, without using WLOG.