Convergence of sequence of real function

Question

We know that if a sequence is monotonically increasing and bounded above then it is convergent. But, in case f:R->R be strictly increasing continuous function . If $a_ {n}$ is a sequence in [0,1] , then the sequence ${f(a_{n})}$ is bounded and Since , f a function on [0,1] and c is monotonically increasing then should attain its infimum at {f(0)} and supermum at {f(1)}. Therefore sequence ${f(a_{n})}$ should converge to f(1). Please let me know about convergence of f