Show that a set of homotopy classes has a single element This is from Munkres section 51 problem 2b

Given spaces $X$ and $Y$ let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$.  Show that if $Y$ is path connected, the set $[I,Y]$ has a single element.  Here $I=[0,1]$.

My approach to the problem is as follows.  Let $f,g \in [I,Y]$ and $f,g:I \to Y$ be continuous functions.  First I define a function $F(x,t)=f(x(1-t))$.  Hence $F(x,0)=f(x)$ and $F(x,1)=f(0)$.  This show that $F$ is a homotopy from  $f$ to $e_{f(0)}$.  Likewise, define $G(x,t)=g(xt).$  Here $G(x,1)=g(x)$ and $G(x,0)=g(0)$.  This shows that $G$ is a homotopy from $g$ to the constant function $e_{g(0)}$.  Since $Y$ is path connected we can find a path that forces a homotopy between $e_{g(0)}$ and $e_{f(0)}$.  Hence $f\cong e_{f(0)}\cong e_{g(0)} \cong g$.
Is this the right strategy?

Your proof is correct!
Just one comment, though: perhaps you can mention why the maps $F$ and $G$ are continuous, for the sake of more clarity.
