Find a subgame perfect equilibrium and a Nash equilibrium
As afreelunch only answered in the comment section, I'll answer it here.

The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call $S$ the set of all SPNE and $N$ the set of all NE in a game. Then:

$$S \subseteq N.$$

If the sequential game has only one subgame (see the subgame definition here), the game itself, then we will have. 

$$S = N.$$

And my reasoning was right.