Are hydrogen spectral differences guaranteed to be unique? I was shown an interesting question that goes something like this:

Consider the energy released by the transition in Hydrogen when the electron falls from $n=4$ to $n=3$. Explain how we know that this could not have been released from another transition e.g. from $n=100$ to $n=10$ etc.

I thought that it could be possible that at least one of the other transitions in the atom could happen to coincide with the energy levels of another.
As long as the fall from $n=4$ to $n=3$ is not greater than 50% of the total ionization energy (i.e. $n=\infty$ to $n=1$), then shouldn't it be possible that another transition matches it's energy? Considering how as $n$ approaches infinity, the energy level differences approach 0, this seems likely since we have such fine control.
 A: We need to find solutions for $n_1$ and $n_2$ such that $\frac{1}{n_1^2}-\frac{1}{n_2^2} = \frac{1}{9}-\frac{1}{16} = \frac{7}{144}$, here $n_1$ is the lower orbit and $n_2$ is the upper orbit. Note that when $n_1 >=5$, $n_2^2 <0$ hence there is no solution for $n_2$ in this case. So the lower orbit to which the electron makes transition to should be less than 5. Now for each $n_1 <5$ we can try to find $n_2$ satisfying the equation, $n_1 = 3$ and $n_2 = 4$ will come out to be the only solution.

Considering how as n approaches infinity, the energy level differences approach 0, this seems likely since we have such fine control.

As for n tending to infinity $\frac{1}{n^2}$ tends to zero hence it might seem that varying $n_2$ at very large value does not changes the energy released by much amount so we might be able to find $n_1$ and $n_2$ such that the energy released comes out to be approximately equal to that for a transition from n = 4 to n=3, but we showed that even for $n_2 = \infty$, $n_1$ should be <5. Here variying $n_2$ might not affect the energy released by much and we do have a smooth control on the values of energy released but to get that value closer to the required one we need to vary $n_1$ which is <5 and for which energy released value changes by large amounts.
