induction - Proof on String Reverse

I'm having trouble with a proof on a string reversal if anyone could lend a hand.

Given the recursive definition of String Reverse, R:

• $\varepsilon^R = \varepsilon$




• $(ax)^R=x^Ra, for x\in\sum^*$

Prove that $(xa)^R=ax^R$

My first instinct was to use proof by induction.

(Base Case) $|x|=0, i.e. x=\varepsilon$

LHS: $(xa)^R=(\varepsilon a)^R=(a)^r=a$

RHS: $a\varepsilon^R=a\varepsilon=a$

So our base case holds.

(Inductive Hypothesis) Assume true for $|x|=k, k \ge 0$,

$(xa)^R=ax^R$

(Inductive Step)

This is where I am stuck, as I know I have to show for the k+1 case, I can not figure out where to go from here. Any help would be appreciated, thanks!