Recall that a coherent sheaf $\mathcal F$ on $\mathbb P^n$ is $m$-regular if $H^i(\mathbb P^n,\mathcal F(m-i)) = 0$ for all $i\geq 1$. Here's an exercise from Jarod Alper's marvelous notes on Stacks and Moduli.
Show that if $\cdots\to\mathcal F_2\to\mathcal F_1\to\mathcal F_0\to\mathcal F\to 0$ is a resolution of coherent sheaves with $\mathcal F_i$ an $m+i$-regular sheaf for all $i$, then $\mathcal F$ is $m$-regular.
My idea was to break up the resolution into short exact sequences $0\to \mathcal K_0\to\mathcal F_0\to\mathcal F\to 0$ and $0\to\mathcal K_{p}\to\mathcal F_{p}\to\mathcal K_{p-1}\to 0$, take twists and do a computation with the associated long exact sequences. For instance, twisting the first sequence by $m-i$ for $i\geq 1$, I get a portion of the LES in cohomology which reads (suppressing $\mathbb P^n$) $$H^{i-1}(\mathcal F(m-i))\to H^i(\mathcal K_0(m-i))\to H^i(\mathcal F_0(m-i))\to H^i(\mathcal F(m-i))\to H^{i+1}(\mathcal K_0(m-i)),$$ and we know that $H^i(\mathcal F_0(m-i))$ vanishes, so $$H^{i-1}(\mathcal F(m-i))\to H^i(\mathcal K_0(m-i))\to 0$$ and $$0\to H^i(\mathcal F(m-i))\to H^{i+1}(\mathcal K_0(m-i))$$ are exact. The second sequence is probably more useful, since it should be enough to show that $H^{i+1}(\mathcal K_0(m-i)) = 0$.
Well anyway, continuing, I have exact sequences $$0\to\mathcal K_{p}\to\mathcal F_{p}\to\mathcal K_{p-1}\to 0$$ for all $p\geq 1$. Twisting by $m+p-i$ and taking cohomology, I get $$\cdots\to H^{i-1}(\mathcal K_{p-1}(m+p-i))\to H^i(\mathcal K_p(m+p-i))\to H^i(\mathcal F_p(m+p-i))\to H^i(\mathcal K_{p-1}(m+p-i))\to H^{i+1}(\mathcal K_{p}(m_p-i))\to\cdots$$ and since $H^i(\mathcal F_p(m+p-i)) = 0$ for all $i\geq 1$, I get exact sequences $$H^{i-1}(\mathcal K_{p-1}(m+p-i))\to H^i(\mathcal K_p(m+p-i))\to 0$$ and $$0\to H^i(\mathcal K_{p-1}(m+p-i))\to H^{i+1}(\mathcal K_{p}(m+p-i)).$$ Replacing $i$ with $i+p$, the second sequences becomes $$0\to H^{i+p}(\mathcal K_{p-1}(m-i))\to H^{i+p+1}(\mathcal K_{p}(m-i)).$$ From here I get a sequence of inclusions $$H^i(\mathcal F(m-i))\subset H^{i+1}(\mathcal K_0(m-i))\subset\cdots\subset H^{i+p}(\mathcal K_p(m-i))\subset\cdots.$$ But I know nothing about the regularity of $\mathcal K_p$. How do I conclude?
I'm an idiot. $H^{i+p}(\mathbb P^n,\mathcal K_p(m-i)) = 0$ as soon as $i+p>n$.