Prove there exists a point $c$ such thst $f(c)=c$ for the following function 
If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function with $f(0)=2$ and $|f'(x)| \leq 1/2$ for all $x$ then there is a point $c$ such that $f(c)=c$ .  

My Attempt 
Let $$h(x)=f(x)-x$$ Now at the point $x=0$ $h(0)=f(0)-0=2$ now we need to find a point $x_0$ such that $h(x_0)<0$ and then we can apply the intermediate value theorem,.
Take the interval $(0,5)$ by the mean value theorem there exists a $c \in (0,5)$ such that $$f'(c)=\dfrac{f(5)-f(0)}{5-0}=\dfrac{f(5)-2}{5}$$
We know from the question that $$\left|\dfrac{f(5)-2}{5}\right| \leq 1/2$$ and so solving gives us $$-1/2 \leq f(5) \leq 4.5$$ and so for any value in that range $$h(5)=f(5)-5 <0$$ and by intermediate value theorem there exists a $c \in (0,5)$ such that $h(c)=0 \iff f(c)=c$
Is this correct? Any help is much appreciated.

Yes it is. By any chance though, are you allowed to apply the fixed point theorem? Also though, to apply Bolzano/Intermediate Value you should justify continuity. This is a little nitpicky and is a question of two or three words, but still it's nice to include it.
