$\gdef\CC{\mathbb{C}}$
Let $D$ be an open disk in $\CC$. Suppose give two holomorphic functions $f,g: D \to \CC$, which agree on some smaller open disk $D' \subset D$. Then $f = g$ on all of $D$.
Let $r$ denote radius of $D'$. Consider the points along the line segment joining the center of $D'$ to that of $D$ which are evenly spaced at distance $r/2$ apart, starting with the center of $D'$; label them $z_0, z_1, \cdots, z_n$. Since $z_1$ lies in $D'$, it follows that the derivatives of $f$ and $g$ to all orders agree at $z_1$, Therefore the Taylor expansions of $f$ and $g$ agree at $z_1$. Now recall a theorem from last lecture: the Taylor series expansion of a function converges to that function wherever that function is defined and holomorphic. It follows in particular that $f$ and $g$ agree on the disc $D'_1$ of radius $r$ centered at $z_1$. Continuing in this manner, we find inductively that $f$ and $g$ agree on all the analogous disks $D_i'$, and in particular agree on $D_n'$. But $D_n'$ contains the center of $D$; thus $f$ and $g$ have the same Taylor expansions at the center of $D$, and hence by the same theorem agree on all of $D$.
There is an extension of this theorem, where $D$ is replaced by a more general open subset of $\CC$. The relevant notion is connectedness.