Bsrgvty

The Prime Number Theorem.

Chebyshev proved that if $$lim_ntoinftypi(n)log nover n$$ exists (here, $pi(n)$ is the number of primes up to $n$), then it equals $1$. Fifty years passed after that before Hadamard and de la Vallée Poussin (independently) proved that the limit exists.

Isoperimetric problem. Using clever geometric argument Steiner proved in 1838 that if there is a geometric figure of the highest area for a given perimeter, it must be a circle. However, it was only Blaschke in 1919 who showed the existence of such a figure.

By the way, Aknazar Kazhymurat's answer is approximately the Perron's joke about invalidity of Steiner's proof.

User bof, in a comment, pointed to the history of The Monster. I think that's an excellent example, and worth expanding on. I quote from Wikipedia:

The Monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess (1976) as a simple group containing a double cover of Fischer's Baby Monster group as a centralizer of an involution. Within a few months, the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the Monster actually exists. [Emphasis mine] Griess (1982) constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative algebra; he first announced his construction in Ann Arbor on January 14, 1980.... Griess's construction showed that the Monster exists.

Either Tauber's original theorem or Littlewood's improvement upon it is of this form. Let $a_n$ be a sequence of real numbers which is $o(1/n)$ (for Tauber's version) or $O(1/n)$ (for Littlewood's verion) and suppose that $lim_x to 1^- sum_n=1^infty a_n x^n = L$. Then $sum_n=1^infty a_n = L$.

The hard part is that $sum_n=1^infty a_n$ converges. Assuming the sum converges, the easier Abel's theorem tells us is must converge to $L$.

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

Simple examples are given by limits of several variables (because we can often find a path along which the limit is trivial). For example, assuming the existence of
$$lim_(x,y)to(0,0)fracxy^2+sin(x)x^2x^2+y^2qquadtextandqquad lim_(x,y,z)to(0,0,0) (x^2+y^2+z^2)^x^2y^2z^2,$$
we obtain
$$lim_(x,y)to(0,0)underbracefracxy^2+sin(x)x^2x^2+y^2_:=f(x,y)=lim_yto 0 f(0,y)=lim_yto 0 0=0$$
and
$$lim_(x,y,z)to(0,0,0) underbrace(x^2+y^2+z^2)^x^2y^2z^2_:=g(x,y,z)=lim_xto 0 g(x,0,0)=lim_xto 0 1=1.$$

If $0<x$ is fixed, define the sequence $x_n$ recursively by $x_n+1=x^x_n$ and $x_0=x$. It is easy to show that if the limit $l$ of the sequence $x_n$ exists, then $(frac1e)^eleq x leq e^frac1e$ and $x=l^frac1l$. I found it much tougher to show that in that range of $x$, the limit does indeed exist. (This is an exercise in Knopp's book on infinite series).

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

In the theory of percolation and other statistical physics models such as self-avoiding walks, it is common to encounter so-called critical exponents, which are rational numbers that describe a fractional exponent related to the rate of decay of a particular function, often a probability, as a parameter tends to some value.

One example of this is the exponent 5/48, tied to the asymptotic relation
$$ pi(n) = n^-5/48+o(1) qquad (ntoinfty), $$
where $pi(n)$ denotes the probability that the connected component that contains the origin in critical percolation over a two-dimensional lattice has diameter at least $n$.

Another example is the exponent 5/36, which appears in the formula
$$ theta(p) = (p-p_c)^5/36+o(1) qquad (psearrow p_c). $$
Here, $p_c$ denotes the critical percolation probability, and $theta(p)$ is the probability that the origin belongs to the infinite percolation cluster. See Wendelin Werner’s notes, pages 3-4 for these two examples. See this Wikipedia page for many others.

The interesting thing about the critical exponents is that although their values are in many cases known precisely, they have been computed using mostly nonrigorous methods. However, as far as existence goes, they are conjectured to exist (in the sense that the relevant asymptotic relations such as the ones written above hold) in fairly large generality for any “reasonable” lattice, and the actual existence has either not been proved at all (I believe that’s the case for most or all exponents related to self-avoiding random walks), or has been proved only for very specific lattices (using SLE techniques pioneered by Schramm, Lawler, Werner, Smirnov and others). The above two formulas were proved in the case of the triangular lattice, as Werner explains in his notes.

Let's take the expression "the largest positive integer". Assuming existence, we get that $n^2leq n$, but we also have $n^2geq n$, so $n^2=n$, i.e. $n=1$. Proving existence seems to be harder: can you prove false statements in ZFC (or prove that you can't, for that matter)?

There are a number of examples from number theory of finite subsets of $mathbbZ$ whose largest element is unknown. For instance, let $S$ be the set of all positive integers that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers. It is known that $|S|leq 19$, and eighteen elements of $S$ are known, the largest being 462. If there is another element $n$, then $n>10^11$. Assuming the Generalized Riemann Hypothesis, $n$ does not exist. See http://people.math.sfu.ca/~kkchoi/paper/rep.pdf.

Minimal surfaces would provide a lot of examples.

Construction of minimal surfaces often reduces to finding a meromorphic and a holomorphic functions (Weierstrass representation). Often we have a general idea about these functions up to some parameters, which are determined by "closing the periods", meaning that some path integrals should vanish.

In many cases the periods are easily closed numerically, leading to beautiful pictures, and the surface is naturally named after the discoverer (they deserve). But existence of period-closing parameters could be very hard prove, sometimes never done. The Horgan (non)surface is a famous example that computer closes periods which is later proved impossible. Then there is the embeddedness to prove, which could be even harder.

It may be hard to prove that a given odd function $f:mathbbRtomathbbR$ is integrable, even measurable. However, if it is integrable, $int_mathbbRf(x)dx=0$. (This example is deliberately trivial).