Contraction mapping in the context of $f(x_n)=x_{n+1}$.

I'm interested in convergence of $f(x_n)=x_{n+1}$ and often hear this term referenced.

What does it mean to be a contraction mapping in the context of the sequence of real numbers given by $f(x_n)=x_{n+1}$? And what does it tell us about such sequence?

A search online gives the answer given by Wikipedia, that it is a function $f$ defined on a metric space $(M,d)$ from $M$ to itself with the property that for some real number $k \in [0,1)$,

$$d(f(x),f(y)) \leq k d(x,y)$$

This definition is quite unhelpful because I barely know what a metric space is. I feel like this definition is to generalized, and I'm interested in in the specific case of recursively defined real sequences $f(x_n)=x_{n+1}$ that there should be a more specific answer to my question.

Contraction maps should be considered together with Banach fixed-point theorem. So ...

What does it mean to be a contraction mapping in the context of the sequence of real numbers given by $f(x_n)=x_{n+1}$?

As others stated $d(x,y)= |x-y|$ is a metric.

And what does it tell us about such sequence?

Well, it tells that the sequence is Cauchy. I.e. $$|x_{n+1}-x_{n}|=|f(x_{n})-f(x_{n-1})|\leq k |x_{n}-x_{n-1}|\leq ... \leq k^{n}|x_1-x_0|$$ and with this in mind: $$|x_{n+p}-x_n|=|x_{n+p}-x_{n+p-1}+x_{n+p-1}-x_n|=|x_{n+p}-x_{n+p-1}+...+x_{n+1}-x_n|\leq \\ |x_{n+p}-x_{n+p-1}|+...+|x_{n+1}-x_n| \leq k^{n+p-1}|x_{1}-x_{0}|+..+k^{n}|x_{1}-x_{0}|=\\ |x_{1}-x_{0}|\left(k^{n+p-1} + ... +k^{n}\right)=k^{n}|x_{1}-x_{0}|\left(k^{p-1} + ... +1\right)<...$$ with the last term, we can add all the terms of the geometric progression because $0\leq k < 1$ and $$... < k^{n}|x_{1}-x_{0}|\sum_{t=0}^{\infty}k^t=\frac{k^n}{1-k}|x_{1}-x_{0}| \rightarrow 0, n\rightarrow \infty$$

But, we know that every Cauchy sequence in a complete metric space has a limit and every compact is also complete. Also, every segment of the form $[\alpha, \beta] \subset \mathbb{R}$ is compact.

Here comes Banach fixed-point theorem saying that if $f:[\alpha, \beta] \rightarrow [\alpha, \beta]$ is a contraction mapping then it admits a unique fixed point $f(x^*)=x^*$ and for $\forall x_0 \in [\alpha, \beta]$ the sequence constructed like $x_{n+1}=f(x_n)$ has $x^*$ as its limit. From this perspective

Contraction mapping in the context of $f(x_n)=x_{n+1}$.

is a way to check if the sequence converges.

The usual metric on the real numbers is $$d(x,y) = |x-y|.$$

In other words, we have that $$d(f(x),f(y))\leq k\cdot d(x,y)\quad\implies\quad |f(x)-f(y)|\leq k\cdot |x-y|.$$

If $f$ is a function from $\Bbb R$ to $\Bbb R$ (that is, its input and output is a real number), then $f$ is a contraction mapping if there is a number $0 < k < 1$ for which $$ |f(x) - f(y)| \leq k|x - y| $$ for all $x,y$.