Identity Mapping to Coarser Topology is Continuous

Theorem
Let $S$ be a set.

Let $\tau_1$ and $\tau_2$ be topologies on $S$.

That is, let $T_1 = \struct {S, \tau_1}$ and $T_2 = \struct {S, \tau_2}$ be topological spaces.

Let $I_S: \struct {S, \tau_1} \to \struct {S, \tau_2}$ denote the identity mapping on $S$:
 * $\forall x \in S: \map {I_S} x = x$

Then:
 * $I_S: T_1 \to T_2$ is a continuous mapping


 * $\tau_2$ is coarser than $\tau_1$.
 * $\tau_2$ is coarser than $\tau_1$.

Proof
But $\tau_2 \subseteq \tau_1$ is the definition of $\tau_2$ being coarser than $\tau_1$.