Can I say range $T = $ dim null $S$ + range $S$ if $S \in \mathcal{L}(V,W)$, and $T \in \mathcal{L}(U,V)$?

Can I say range $ T = $ dim null $ S$ + range $ S$ if $ S \in \mathcal{L}(V,W)$ , and $ T \in \mathcal{L}(U,V)$ ?

The reason I asked this is because I am doing Q22 of section 3.B on Linear Algebra Done Right.

Suppose $ U$ and $ V$ are finite-dimensional vector spaces and $ S \in \mathcal{L}(V,W)$ , and $ T \in \mathcal{L}(U,V)$ . Prove that $ $ \text{dim null }ST \leq \text{dim null }S + \text{dim null }T$ $

My failed trail is as follow:

range $ T = $ dim null $ S$ + range $ S$ (from S)

dim $ U$ = dim null $ T$ + range $ T$ (from T)

dim $ U$ = dim null $ ST$ + range $ ST$ (from $ ST \in \mathcal{L}(U,W)$ )

then

dim null $ T$ + range $ T$ = dim null $ ST$ + range $ ST$

dim null $ T$ + dim null $ S$ + range $ S$ = dim null $ ST$ + range $ ST$

dim null $ ST$ = dim null $ T$ + dim null $ S$ + range $ S$ – range $ ST$

The trial seems not working. Maybe range $ T = $ dim null $ S$ + range $ S$ is wrong in the first place.