Let $f$ be an isometry of $\mathbb{R}^n$.
And, let's say $f$ satisfies $ \forall x \in \mathbb{R}^n$, $\text{d}(f(x),x) < K$ for some $K > 0$.
My attemtpt:
Every isometry could be expressed as $t\circ g$ where $t$ is a translation, and $g$ is an orthogonal transformation.
So, let $f(x) = t\circ g (x)$, and see what happens if $g \neq I$.
By assuming $g \neq I$, we can get a line where $g(x)$ and $x$ belongs to.
Now, by $t$, we will get $t(g(x))=g(x)+a$ for some $a \in \mathbb{R}^n$.
I also applied $g$ once again to $g(x)$.
At this point, I tried to draw each point and tried to use the triangle inequality to intuitively see the relationships between points (when $g$ is not identity.) However, I couldn't get any useful information from the drawing. Also, it made me even more confused.
What should I consider for this problem?
Edit:
I've been trying to see what gives me a contradiction by letting $a=0$ as "Moishe Kohan" suggested.
But, I'm still stuck. Using the triangle inequality didn't really help. Drawing also didn't.
However, there's one I found, which is $|a| < K$. I got it from plugging in $x=0$. (I'm not sure if this would be useful though.)
May I have more hints to go further?