Proving $dim(img(g\circ f)) \leq dim(im(f))$ I want to prove: $dim(img(g\circ f)) \leq dim(im(f))$. $g$ and $f$ are linear maps.
The map $f$ first maps the input $x$ into $im(f)$. $g$ take $im(f)$ into $im(g)$. Thus, this last map is $g_{im(f)}$, $g$ restricted to image of $f$.

For any linear map $L$, we must(*) have $dim(img(L))\le dim(dom(L))$, i.e. the image has dimension no larger than the domain.
Hence $dim(img(g_{im(f)}))\le dim(dom(g_{im(f)}))=dim(im(f))$.
(*) This follows from the Rank-Nullity Theorem, i.e. $$dim(im(L))+dim(ker(L))=dim(dom(L))$$
