Inverse triangle inequality on $\mathbb{R}^n$ I do know the basic triangle inequality on $\mathbb{R}^n$, however I am not really sure how I can solve the problem below.
$\|x_n - x\| \leq 1  \Rightarrow   \|x_n\| \leq \|x\| + 1$ (By the Triangle Inequality)
Edit: all the answers are very good I flagged Valerin because of the simplicity.
 A: Actually if you are familiar with inverse triangle inequality, you are done!
$$|\|x_n\|-\|x\||\leq \|x_n-x\| \leq 1$$ then $$\|x_n\|-\|x\|\leq 1\Rightarrow \|x_n\|\leq \|x\|+1$$
A: $||x_n -x|| \le 1$
$||x_n -x|| + ||x|| \le 1+||x|| $
$||(x_n -x)+x|| \le||x_n -x|| + ||x|| \le ||x||+1$
$||x_n||\le ||x||+1$
....
Negative signs always throw me.
But if you $a = m-n$, and $m = a+n$ you can always show the "flipside" to the triangle inequality

Corollary: $||m - n|| \ge ||m|| - ||n||$

Pf:  $||a +n|| \le ||a|| + ||n||$
$||m|| \le ||m-n|| + ||n||$.
$||m||-||n|| \le ||m-n||$.
That always throws me!
... but if if it didn't ....
$||x_n -x|| \le 1\implies$
$||x_n|| - ||x|| \le ||x_n - x|| \le 1\implies$
$||x_n|| \le 1 + ||x||$ would follow easily.
A: Here it is a proof related to the result in question.
Based on the triangle inequality, notice that
\begin{align*}
\|x_{n}\| = \|x_{n} - x + x\| \leq \|x_{n} - x\| + \|x\| \Rightarrow \|x_{n}\| - \|x\| \leq \|x_{n} - x\|
\end{align*}
At your case, we have
\begin{align*}
\|x_{n}\| - \|x\| \leq \|x_{n} - x\| \leq 1 \Rightarrow \|x_{n}\| \leq \|x\| + 1
\end{align*}
and we are done.
Hopefully this helps!
A: The basic form of the triangle equality is
\begin{equation}
||a+b|| \leqq ||a||+||b||.
\end{equation}
If you replace $a$ for $a-b$, you get
\begin{equation}
||a||-||b||\leqq ||a-b|| \cdots \ast
\end{equation}
If you use $\ast$ for $a=x_n, b=x$, you can see $||x_n||-||x||\leqq ||x_n-x||$ .  And now, $||x_n-x||\leqq 1$ holds. Therefore, $||x_n||-||x||\leqq 1$. So $||x_n||\leqq ||x|| +1$.
