Let $\theta$ be an irrational number with continued fraction expansion $[a_0; a_1, a_2, \cdots]$.
Suppose $P_n/Q_n = [a_0; a_1, \cdots , a_n]$ is the $n^{th}$ convergent. Then how do I show that $P_0=a_0$.
I have that $P_0/Q_0= [a_0]$ but where do I go from here?
As you observed, the $n$-th convergent of the continued fraction $[a_0; a_1,a_2,\dotsc]$ is the rational number $$ c_n = [a_0; a_1,\dotsc,a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}} $$ and the integers $p_n,q_n$ are simply numerator and denominator of $c_n$, respectively.
In particular, $c_0 = a_0 \in \Bbb{Z}$ has denominator $1$, so $p_0 = c_0$.