$a_n$ be a sequence in [0,1]. what about f($a_n$)? consider a strictly increasing function f. let $a_n$ be a sequence in [0,1]. which of following is true about f($a_n$)?


*

*f($a_n$)  is bounded 

*f($a_n$)  is convergent

*f($a_n$)  is bounded but not convergent

*f($a_n$)  is unbounded
my attempt:
let f=x, so f($a_n$)=$a_n$. since nothing can be surely said about an except that it is bounded. so f($a_n$) is bounded. plz give your suggestion

Assuming that we are dealing with a strictly increasing function $f:[0,1]\rightarrow\mathbb R$ we conclude that $f(0)\leq f(a_n)\leq f(1)$ for each $n$. 
This tells us that the sequence $(f(a_n))_n$ is a bounded sequence. 
The sequence can be convergent, but that is not necessary. E.g. let $f$ be prescribed by $x\mapsto x$. Then $f(a_n)=a_n$ and the sequence $(f(a_n))_n$ iff sequence $(a_n)_n$ is convergent.
