The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+...

Given: A natural $N$ and the finite geometric identity used below.

A series converges exactly when its sequence of finite partial sums converges (Series, partial sums, convergence and the sum, divergence, and the tail series).

The Riemann integral is linear over finite sums (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$).

The power rule gives $(x^{m+1}{)}^{\mathrm{\prime }}=(m+1)x^m$ for $m\ge 0$. If $G$ is differentiable on a closed interval and $f=G^{\mathrm{\prime }}$ is integrable there, then $\int f$ is the endpoint increment of $G$ (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$).

If $m\le f\le M$ on $[a,b]$, then the integral lies between $m(b-a)$ and $M(b-a)$ (If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$).

Substitution holds for a differentiable inner map with integrable derivative and a continuous outer function (Substitution: if $\phi$ is differentiable on $[c,d]$ with ${\phi }^{\mathrm{\prime }}$ integrable and $f$ is continuous on an interval containing $\phi ([c,d])$, then ${\int }_{\phi (c)}^{\phi (d)}f={\int }_c^d(f\circ \phi ){\phi }^{\mathrm{\prime }}$).

On its natural domain, $\tan t=\sin t\mathrm{/}\cos t$, $\sec t=1\mathrm{/}\cos t$, and $(\tan t{)}^{\mathrm{\prime }}={\sec }^{2}t$; for every real $t$, ${\sin }^{2}t+{\cos }^{2}t=1$ (Tangent, cotangent, secant, and cosecant on their exact natural domains, Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant, Parity and the Pythagorean identity for sine and cosine).

The quarter-turn values are $\sin (\pi \mathrm{/}2)=1$ and $\cos (\pi \mathrm{/}2)=0$; the sine and cosine addition formulas hold for all real inputs, sine is strictly increasing on $[-\pi \mathrm{/}2,\pi \mathrm{/}2]$, cosine is strictly decreasing on $[0,\pi ]$, and $\sin 0=0$, $\cos 0=1$ (Quarter-turn values and shifts by pi/2 and pi, The addition formulas for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, The derivatives of sine and cosine are cosine and minus sine).

A sequence squeezed between two sequences with the same limit has that limit (The squeeze theorem).

For every $\epsilon >0$ there is a natural $M\ge 1$ with $1\mathrm{/}M<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).