Let $F:X\to Y$ and $G:Y\to Z$ be two linear maps between vector spaces.
How do you show that $$ \dim(\mathrm{coker}(G\circ F))) \leq \dim(\mathrm{coker}(F))+\dim(\mathrm{coker}(G)) $$?
It is clear that $\mathrm{im}(G\circ F) \subseteq \mathrm{im}(G)$ because $F(X)\subseteq Y$. This gives the intuitive notion that $\mathrm{coker}(G\circ F) \supseteq \mathrm{coker}(G)$ (morally), and so $\dim(\mathrm{coker}(G))\leq\dim(\mathrm{coker}(G\circ F)$. How to proceed from here?
There is this answer: https://math.stackexchange.com/a/1113704/61151 but it doesn't go into enough detail (e.g. how does one get the S.E.S. and how to infer from them a formula on the dimensions).