There are a lot of proofs whose clever simplicitly is worth writing down.

"... the snobbery of the young, who think that a theorem is trivial because its proof is trivial." -- 'the quote is a garbled version of Grothendieck, quoting Ronnie Brown quoting J.H.C. Whitehead. I found it on p.188 of the PDF version of Récoltes et Semailles'

• Lagrange's Theorem
• Bayes' Theorem
• Louiville's Theorem (the complex one)
• Uncountability of the reals
• Infinitely many prime numbers exist
• Union bound on probability
• Markov's Inequality
• Chebyshev's Inequality
• The theorem that there are two irrational numbers a and b with ab rational
• Poincaire Recurrence Theorem
• q-norm <= p-norm if \(p < q\)
• If in a list of \(n\) vectors, each is within \(\frac{1}{\sqrt{n}}\) of an orthogonal basis, then your vectors are independent.
• Proof of linear independence of \(e^{a t}\).
• Double counting proof that \(\sum_{i=0}^n \binom{n}{i}^2 = \binom{2 n}{n}\).
• Double counting proof that \(\sum_{i=0}^j (-1)^i \binom{n}{i} = (-1)^j \binom{n-1}{j}\).
• Double counting proof that \(\sum_{k=0}^n k \binom{n}{i}^2 = n \binom{2 n - 1}{n}\)
• Proof that continuous bijections from a compact space to a hausdorff space is a homeomorphism.
• Irrationality of 21/n for n≥3: if 21/n=p/q then pn=qn+qn, contradicting Fermat's Last Theorem. Unfortunately FLT is not strong enough to prove 2–√ irrational.
• Proof of Euler Product Formula for the Riemann Zeta function (\(\sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \in \mathrm{primes}} \frac{1}{1 - p^{-s}}\)).
• Centroid proof of \(\cos{\frac{pi}{7}} + \cos{\frac{3 \pi}{7}} + \cos{\frac{5 \pi}{7}} = \frac{1}{2}\).