= Solution
The <first-step analysis> gives $h_0=1$ and, for $k\geq1$,
$$
h_k=p h_{k+1}+q h_{k-1}.
$$
The <minimal nonnegative solution for hitting probabilities> theorem states that the probability of ever hitting a target is the pointwise smallest nonnegative solution of this transition recurrence with value one at the target. It follows by comparing any such solution with the probabilities of hitting within $n$ steps, then taking their increasing limit.
The characteristic equation $pr^2-r+q=0$ has distinct roots $1$ and $\beta=q/p\in(0,1)$. Hence every solution on the nonnegative integers with boundary value one has the form
$$
g_k=A+(1-A)\beta^k.
$$
If it is nonnegative for all $k$, taking $k\to\infty$ forces $A\geq0$, and therefore $g_k-\beta^k=A(1-\beta^k)\geq0$. The candidate $g_k=\beta^k$ is itself nonnegative and satisfies the recurrence and boundary value. It is consequently the minimal solution, giving the <biased random walk hitting probability>
$$
\boxed{h_k=\left(\frac{1-p}{p}\right)^k,\qquad k>0;\qquad
\beta=\frac{1-p}{p}.}
$$
On the negative half-line the already established values $h_k=1$ satisfy the same recurrence away from zero. No unproved boundary condition at infinity has been imposed.
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