fir - Vector length output of discrete time convolution

Suppose that the impulse response of a discrete time filter is $h[i]$ where $i=0,1,2,...,N-1$ and the input sequence to the filter is $x[i]$ for $i=0,1,2,.., M-1,$ what would be the length of output vector length?

I have concluded it is $N$ because according to discrete time convolution

$$\begin{align} y[0] &= x[0]h[0]\\ y[1] &= x[0]h[1]+x[1]h[0]\\ y[2] &= x[0]h[2]+x[1]h[1]+x[2]h[0]\\ \vdots &=\vdots\\ y[N] &= x[1]h[N-1]+x[2]h[N-2]+...+x[N]h[0] \end{align}$$

Is this correct?