Intersection of Normal groups

Let H and K be normal subgroups of G, prove that H intersect K is also normal in G.

Proof:

Since H and K are normal in G, we have

$\displaystyle aH=Ha \forall a \in G$

$\displaystyle bK=Kb \forall b \in G$

Now, let $\displaystyle c \in H \cap K$, then c must retain the property of H and K since c is in both.

Thus, $\displaystyle a(c)=(c)a \forall a \in G$, implies $\displaystyle a(H \cap K) = (H \cap K)a \forall a \in G $

Therefore H intersect K is normal in G.

Is this right?