If $f$ is an immersion, prove its restriction to any submanifold of its domain is an immersion.
Consider a submanifold $\tilde{X}$ of $X$, and take any point $p \in \tilde{X}$. Then when $d\tilde{f}_p(\tilde{x}) = 0$...
I was not able to show $\tilde{x} = 0$ here. Can I reduce this problem to the canonical submersion? In other words, if $d\tilde{f}_p(\tilde{x}) = 0$, then $df_q(x) = 0$ where $q = (p_1, \dots, p_n, 0 \dots, 0)$, and $x = (\tilde{x}_1, \dots, \tilde{x}_n, 0, \dots, 0$). And because $f$ is an immersion, $df_q(x) = 0$ implies $x=0$ and therefore $\tilde{x} = 0$.
Though, I am not quite comfortable with the interchange between manifold and submanifold. Specifically, I don't know if I can claim if $d\tilde{f}_p(\tilde{x}) = 0$, then $df_q(x) = 0$.
Let $S$ be a submanifold of $M$, and $i : S \to M$ be the inclusion. If $s \in S$, then $di_s : T_sS \to T_sM$ is the inclusion map. As $f|_S = f\circ i$, we have
$$d(f|_S)_s = d(f\circ i)_s = df_{i(s)}\circ di_s = df_s\circ di_s.$$
As $di_s$ and $df_s$ are injective, $d(f|_S)_s$ is injective so $f|_S$ is an immersion.