Abstract algebra: prove that N is a normal subgroup

Let N be a subgroup of a group G. Suppose that, for each x in G, there exists a y in G such that Nx = yN. Prove that N is a normal subgroup.

Thanks for the help!

suppose that for every x in G, there is y in G with Nx = yN.

we would like to show that Nx = xN, for every x, since this would

mean N is normal (if Nx = xN, then xNx^-1 = (Nx)x^-1 = Ne = N).

recall that two left cosets either coincide, or are disjoint.

in particular, x is in just one left coset. since x is obviously in xN

(because x = xe, and e is in N), and x is in Nx = yN, we must have that

xN = yN, so N is normal.