Why is it obvious that attaching an (m+n)-cell to SmVSn is SmxSn?

Also, if you have a space X that is the real line, with two sphere's (S2) wedged at each integer, to determine its homology group, I figured I should use the Mayer–Vietoris sequence, so I split it such that U = X - {odd integers}, V = X - {even integers}. Do you think that's what I should do?