Prove that every continuous map $f: [0,1] \rightarrow [0,1]$ has a fixed point.
Suppose $f$ does not have a fixed point, then $\forall x \in [0,1], f(x) \neq x$.
Thus we have a well defined function $g(x) = \frac{1}{f(x)-x}$. Note that as $g(x)$ is the composition or continuous functions, it must be continuous.
However, $g(0) > 0$ and $g(1) < 0$, so by the intermediate value theorem $\exists x \in [0,1]$ such that $g(x) = 0$. This is clearly impossible.
Thus $f$ has a fixed point.
Best Answer
Your proof is correct. Anyway note that the function $f$ must meet the line $y=x$ in atleast one point, and that point is the fixed point. To see this geometrical idea into a formal proof, set $g(x)=f(x)-x$ and apply IVT to $g$.