Suppose you are trying find an approximation \(u(t)\) to a function \(v(t)\) and you know that the following two equations hold:
\[ u(t_{n+1}) = C(k)u(t_n) \]
\[ v(t_{n+1}) = C(k)v(t_n) + k\tau (t_n) \]
for some smooth function \(\tau (t,k)\) that is not identically 0. Here, \(k > 0\), \(t_n = nk\), and \(C(k)\) is a linear operator which depends on \(k\) but not on \(u(t)\) or \(v(t)\). State and prove conditions on \(C(k)\) and \(\tau (t,k)\) which are sufficient to imply that \[ \lim _{k \to 0,\ nk=t_n=t} u(t_n) = v(t). \]