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Dirichlet's theorem, that if $a,b$ are co-prime natural numbers then $\{an+b: n\in \mathbb N\}$ contains infinitely many primes, was one of the first (if not the first) important result in Number Theory that uses Analysis in the proof.

The irrationality of $\pi$ was shown first by Lambert in the mid-1700's. Niven's remarkably short 20th-century proof is a gem. Both use calculus.

It is known that $e$ and $\pi$ are not Liouville numbers. Perhaps someone who knows something about the proofs could comment on this.

In 2013 a proof was given that, if $p_n$ is the $n$th prime , then $\lim_{m\to \infty}\inf_{n>m}(p_{n+1}-p_n)<7\cdot 10^7.$ Further work has reduced the number from $7\cdot 10^7$ to a value below $300.$ I am not familiar with the proof, but I doubt that it's elementary. No upper bound for $\lim_{m\to \infty}\inf_{n>m}(p_{n+1}-p_n)$ was previously known to exist. Ideally we would like to prove an exact value of $2$ for the $\lim \inf$ (The Twin Prime Conjecture).

Theorem.(Liouville). A bounded entire function $f:\mathbb C\to \mathbb C$ is constant. Proved by considering the formula $f'(z)=(2\pi i)^{-1}\int_{|y-z|=r} f(z)(y-z)^{-2}dy$ as $r\to \infty.$ This theorem can be proved by elementary means by first showing that if $g$ is an entire function then $|g|$ cannot have a non-$0$ local minima, using the power series for $g,$ but that requires showing that $g$ is indeed equal to its power series expansion about any center, which (as far as I know) needs the use of Cauchy-Riemann integral forms.

The Banach-Tarski "Paradox" may qualify. It requires the use of non-Lebesgue sets.

One of Hilbert's famous problem-set was: For $n\geq 2,$ if the polynomial $p:\mathbb R^n\to \mathbb R$ is never negative, is $p$ equal to a finite sum of squares of $rational$ functions from $\mathbb R^n$ to $\mathbb R$? For $n=1$ it is elementary (and simple) that $p$ is a finite sum of squares of polynomials, and there are some simple examples with $n=2$ and $n=3$ where p is not a finite sum of squares of polynomials. Hilbert's problem turns out to be (essentially) the same as showing that the theory of real-closed fields is sub-model complete.

Many conjectures were found to be independent of $ZFC$ by use of Forcing (including Iterated Forcing) and Godel's constructible class $L,$ or were found to be equivalent to, or equi-consistent with, the existence of certain "large cardinals": Examples:

The Suslin (Souslin) conjecture: Can a c.c.c. linear topological space be non-separable ? (Originally, as it is easily seen that an infinite, connected, order-dense, separable linear space without end-points is order-isomorphic to $\mathbb R,$ it was asked whether separable could be weakened to c.c.c.). Arthur Jensen showed that $V=L$ implies "Yes". And Iterated Forcing can show that "No" is equi-consistent with $ZFC.$ Can Lebesgue measure be extended to a countably additive measure whose domain is all sets of reals? This was found (Solovay) to be equi-consistent with the existence of a measurable cardinal.

Finally, I don't know whether there is a "one-dimensional" proof that the value of $J=\int_{-\infty}^{\infty}e^{-x^2}dx$ is $J=\sqrt {\pi}\;$. We have $J^2=$ $\int_{-\infty}^{\infty} e^{-x^2}dx \cdot \int_{-\infty}^{\infty}e^{-y^2}dy=$ $\int_{(x,y)\in \mathbb R^2}(e^{-x^2-y^2})dxdy.$ Now switch to polar co-ordinates.