if $A\subset B\subset C$ prove B is connected if A,C are connected.

Question

$A$ and $C$ are $connected$ subsets of metric space $M$. if $A\subset B\subset C$, prove $B$ is connected.
any idea how to work on this proof.
I am trying to say: let B be disconnected by contradiction .
this mean there exists U, V , subsets of C, such that their union is B and the intersection of the closure of each of them with the other one is empty.(1)
but since B is a subset of C, this implies, U and V are subsets of C , hence subsets of M , with conditions stated in (1). this shows C is not then connected. and this is a contradiction.
I believe something is wrong because I did not use the fact that A is connected or A is a subset of B.
I appreciate any help in strategies to prove connectedness.( even in general )

Answer 1

This is false. Take $A=[-1,1], C=[-2,2]$ and $B=A \cup \{1.5\}$.