Two different numbers never have the same next number. If you know the number after $n$, then you know $n$.

An isolated point of a set $S$ is a point of $S$ that has a neighbourhood with no other point of $S$.

The spring pulls the mass back with a force proportional to the displacement. The mass overshoots the rest position, and the motion repeats forever as a sine wave. A stiffer spring or a lighter mass gives a faster oscillation.

A discrete logarithm of an element $h$ of a cyclic group to a base $g$ is an exponent $k$ such that $g^k = h$. The definition of the discrete logarithm gives the precise form.

At a repeated root, the graph of a real polynomial touches the axis and has a horizontal tangent. So the polynomial and its derivative are both zero there. The algebraic statement is true over every commutative ring.

Every complex number has a square root, a cube root, and so on. For example, $-1$ has the square roots $i$ and $-i$, and $i$ has the square roots $\pm \frac{\sqrt{2}}{2}(1 + i)$. To take an $n$th root, take the $n$th root of the length and divide the angle by $n$.

If $b = f(x)$ with $x$ in both $S$ and $T$, then $b$ is in both images. But $b$ can be in both images for another reason: $b = f(s) = f(t)$ with $s \in S$, $t \in T$ and $s \neq t$.

If all terms eventually stay close to $L$, then the terms that you keep also stay close to $L$. Deleting terms cannot change where the sequence is going.

The Bernoulli distribution with parameter $p$ is the distribution of a random variable with two values. The value $1$ has probability $p$ and the value $0$ has probability $1 - p$. It is the distribution of one Bernoulli trial. The node Bernoulli distribution gives the precise definition.

A circle is the set of all points of a plane at a given distance from a given point. The point is the centre and the distance is the radius. The region inside a circle is a disc.

An increasing sequence that cannot pass a ceiling must pile up somewhere. It piles up at the lowest ceiling, which is the supremum of its terms.

The space of solutions of a linear recurrence is the module of all sequences that satisfy it. It has dimension equal to the order of the recurrence over a field. The node space of solutions of a linear recurrence gives the precise definition.

An essential singularity is an isolated singularity that is neither removable nor a pole, see the definition. Near an essential singularity, a function comes arbitrarily close to every complex value.

A group homomorphism is a map between groups that preserves the group operation. It is the basic way to compare two groups.

The dual map of a linear map $T \colon V \to W$ is the linear map $W^* \to V^*$ that sends each linear form $f$ to $f \circ T$.

The geometric mean of some nonnegative numbers is never larger than their average. For example, for $2$ and $8$, the geometric mean is $\sqrt{16} = 4$ and the average is $5$. Among all rectangles with a fixed perimeter, the square has the largest area. This is the case $n = 2$.

In the discrete topology, every set is open, so each point is isolated from all other points. The integers $\mathbb{Z}$, as a subspace of $\mathbb{R}$, carry the discrete topology.

The subgroups of a quotient group $G/N$ are the same as the subgroups of $G$ that contain $N$. Passing to the quotient does not create new subgroups or lose old ones above $N$. So the picture of the subgroups above $N$ in $G$ is the full picture of the subgroups of $G/N$.

The $n$th roots of unity are the corners of a regular polygon with center $0$. By symmetry, their center of mass is $0$, and so their sum is $0$. For $n = 4$: $1 + i + (-1) + (-i) = 0$.

An alternating series is a series whose terms alternate in sign. Such a series converges when the absolute values of its terms decrease to zero. The node alternating series gives the precise definition.