vector spaces – Prove $dim(im(g circ f)) ≤ min{dim(im g), dim(im f)}$ and $dim(ker(g circ f)) ≥ dim(ker f)$

For the finite-dimensional Vector spaces U, V and W with $f: U → V$ and $g: V → W$ show the following:

  1. dim$($im $(g circ f)) ≤ $min${$dim$($im $g), $dim$($im $f)}$
  2. dim$($ker $(g circ f)) ≥ $dim$($ker $f)$

Im aware of a possible solution to a similar proof to (1) by using the rank nullity problem to show $dim Ker(f+g) leq dim (Ker f cap Ker g) + dim (Im f cap Im g)$, im unsure on how to apply that here though.

For two i got $dim(ker(f circ g)=dim(V)−dim fg(V)≥dim V −dim f(U)=dim(ker f)$
but im not 100% sure about that.