I'm a university student currently studying real analysis. I'm working down my problem sheet and I've been asked the question above. I've had to edit it down because of the character limit so for clarity here is the whole question.
Q: Suppose that $f: [0,1] \rightarrow \mathbb{R}$ is a continuous function with range $[0,1]$. Prove that the equation $f(x) = x$ has at least one solution x in $x \in [0,1]$
I'm still getting the hang of these kinds of questions. I'll consider first the function $g(x) = x - f(x)$. The usual method seems to be, once a function like $g(x)$ is created i.e some function that is a relevant function minus another relevant function, variable or constant, then one says they apply the intermediate value theorem. Is that all you have to do? or are there additional logical steps you need to make a sound argument? Any feedback would be much appreciated!
