TITLE: Mapping on Cauchy Sequences
QUESTION [2 upvotes]: How can I show that $f(x)=x^2$ maps Cauchy sequences to Cauchy sequences?
That is the function preserves the Cauchy property.

REPLY [3 votes]: Let $\left(a_{n}\right)$ be a Cauchy-sequence. Then it is bounded so that $\forall n\;\left|a_{n}\right|<c$
for some constant. Then $\left|a_{n}^{2}-a_{m}^{2}\right|=\left|a_{n}+a_{m}\right|\times\left|a_{n}-a_{m}\right|\leq2c\times\left|a_{n}-a_{m}\right|$.

REPLY [0 votes]: It is a fact that a sequence in $\mathbb R$ is Cauchy iff it is convergent.
Now if $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence then it is a convergent sequence.
Therefore $(f(x_n))_{n\in\mathbb N}=(x_n^2)_{n\in\mathbb N}$ is a convergent and so a Cauchy sequence.
Note that if this is a homework maybe this is not the correct way to prove it. It depents on what you have been taught.
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