Let $\{x_n\}$ be a Cauchy sequence of real numbers, prove that a new sequence $\{y_n\}$, with $y_n$=$x_n^\frac{1}{3}$, is also a Cauchy sequence.
What I am thinking so far is following:
In order to prove $\{y_n\}$ is a Cauchy sequence, we need to find an $N \in \mathbb{N}$ such that for any $m, n \ge N$,
$$
|y_n-y_m| = |x_n^\frac{1}{3}-x_m^\frac{1}{3}| = \frac {\vert{x_n-x_m}\vert} {\vert{x_n^\frac{2}{3}+x_n^\frac{1}{3}x_m^\frac{1}{3}+x_m^\frac{2}{3}}\vert} < \epsilon.
$$
How should I proceed then?
Best Answer
Lemma: For $x, y\in{\mathbb R}$, one has $$|x^{1/3} - y^{1/3}|\le 3 |x - y|^{1/3}$$ Proof
Suppose that $|x|\le |x-y|$. It implies $|y|\le |x| + |y-x|\le 2|x - y|$. Hence $|x^{1/3} - y^{1/3}|\le |x|^{1/3}+ |y|^{1/3}\le (1 + 2^{1/3})|x-y|^{1/3}\le 3 |x-y|^{1/3}$. The same proof holds if one supposes $|y|\le |x-y|$
Suppose now that $|x-y|< \min(|x|, |y|)$. It follows that $x$ and $y$ have the same sign. We may suppose that they are positive. The mean value theorem gives for a $z\in (x, y)$
$$|x^{1/3} - y^{1/3}|\le \frac{1}{3} z^{-2/3}|x - y|\le \frac{1}{3} |x - y|^{-2/3}|x - y|\le \frac{1}{3}|x - y|^{1/3}$$ $\square$
The result is now obvious because for $\epsilon>0$, one can obtain $|x_n^{1/3} - x_m^{1/3}|\le \epsilon$ by imposing $|x_n - x_m|\le\epsilon^3/27$