If $w=a_{1}a_{2}.....a_{n}$ and $x=b_{1}b_{2}....b_{n}$ are strings of the same length, define $alt(w,x)$ to be the string in which the symbols of $w$ and $x$ alternate starting with $w$ that is $a_{1}b_{1}a_{2}b_{2}........a_{n}b_{n}.$If $L$ and $M$ are languages,define $alt(L,M)$ to be the set of strings of the form $alt(w,x),$where $w$ is any string in $L$ and $x$ is any string in $M$ of the same length. Prove that if $L$ and $M$ are regular,so is $alt(L,M).$