Not to rain on any parades but I lived in a walk up -5th floor - and there is NO WAY I could have managed it whilst heavily pregnant and certainly not with an actual baby. Carrying a baby in a pushchair (even a light one) plus nappy bags, shopping, drinks for mum, up and down 4 flights of stairs would be incredibly incredibly hard - especially in the first weeks after birth. I would really recommend having a look to see if you can move somewhere with a lift before you get too big.