In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical:
$$ \sqrt{1+2\sqrt{1+3\sqrt{1+\ldots}}}.
\qquad(\*)
$$
Assuming the above is well-defined, there is a slick trick showing that it [evaluates](https://en.wikipedia.org/wiki/Nested_radical#Ramanujan's_infinite_radicals)
to $3$.


But such careless assumptions can lead to trouble, as in the example of the expression:
$$ -5 + 2(-6 + 2(-7 + 2(-8 + \ldots))).
\qquad(\*\*)
$$


Applying the identity $n = -(n + 2) + 2(n + 1)$
repeatedly for $n=3,4,5,\ldots$, we get
\begin{align}
3 &= -5 + 2(4) \\
 &= -5 + 2(-6 + 2(5))\\
 &= -5 + 2(-6 + 2(-7 + 2(6))\\
 &= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\\
 &=\ldots,
\end{align}
which would falsely suggest that $(\*\*)$ evaluates to $3$.


What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?


Edit: clarifying in light of some of the discussion in the comments. I can see how $(\*\*)$ can also invite examples of false conclusions from an assumption of existence. That was not the intent of the question; the sole point of $(\*\*)$ was to show that the solution technique to $(\*)$ provided at the link can in general yield false conclusions if existence is assumed without additional proof. The spirit of the question is to exhibit cases where the limit exists, but its value given the existence is much easier to establish than the existence itsef.



Examples where existence is harder than evaluation
Brownian motion is an example of this phenomenon in probability.


I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$. (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on.


But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923). 


(From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.)


(The historical notes are from [Pitman and Yor, *Guide to Brownian Motion*](https://arxiv.org/abs/1802.09679), which see for more references.)


