The Kakeya conjecture posits that any [Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture) in $\mathbb{R}^n$ has dimension $n$. 


A discrete (finitized?) version of this problem is the [Finite Field Kakeya conjecture](https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/), which was [proved](https://arxiv.org/abs/0803.2336) by Dvir in 2008.


My understanding is that the Finite Field Kakeya Conjecture was proposed at the end of the twentieth century with the hope that it would lead to methods that could be applied to the original Kakeya conjecture. 
However, it seems that in this case the approach used for resolving the discrete analogue is not easily applied to the original problem, so that the Kakeya conjecture still remains open.


My question is asking for examples of problems where discretizing "succeeded."


**Question:**
Are there examples of math problems where looking at a finite or discrete variant of the initial statement *did* lead to a solution (or if not a complete solution, at least significant progress) of the original problem?
If so, what are they? 


**Edit:**


As commenters pointed out, this question is similar to the one [here](https://mathoverflow.net/questions/270930/when-has-discrete-understanding-preceded-continuous) which requests examples where a discrete version of a theory was developed before the continuous version of the same theory.


My question is different from that previous question, in that I am not interested in cases where discrete problems predated continuous problems.
Instead, I'd like to learn about instances where a continuous problem was *already* proposed, and studying a discrete version of that problem helped inform a solution to the continuous variant. 



Examples of problems where considering "discrete analogues" has provided insight or led to a solution of the original problem
As a non-expert, I will tell a story (since I am not qualified to do more):


Once upon a time (1859) there was conjecture about the zeros of a complex function known as the [Riemann Hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis) (RH). 


Years later Weil formulated a "discrete version" (in terms of finite fields $\mathbb{F}\_q$) of this conjecture; the third of his [Weil Conjectures](https://en.wikipedia.org/wiki/Weil_conjectures) (1949). In 1974, [Deligne](https://en.wikipedia.org/wiki/Pierre_Deligne) proved this discrete version to much acclaim. Here is a well regarded [exposition](http://jmilne.org/math/xnotes/pRH.html) by Milne.


Years later still, Connes is motivated to explore [a potential proof](https://arxiv.org/abs/1509.05576) of the original RH along the lines of Deligne's proof by taking a "limit as $q\to 1$".


These attempts are probably uncontroversially considered mathematical advances. Some, presumably Connes himself, consider this work an advance in the problem at hand (and so making this hopefully an appropriate response to this MO question). However, there are [other opinions](https://mathoverflow.net/q/259263/12218), and so this MO post might prove controversial too.


