It's fashionable to claim that social mobility is dead and "inequality" rules.
Inequality and social mobility are two different things. The first refers mainly to income - inequality simply refers to the gap between high earners and low earners, and thanks to the presence of the City of London and the high earners there, there is high inequality in the UK. Though this is mainly because those at the top earn a lot, rather than those at the bottom getting poorer. All indicators are that those at the bottom in the UK are much better off than they were 20 years ago, with many owning their own homes.
Social mobility is a whole other thing. It's the ability to move up into a different class than you were born into - so your children are a higher class than you are. Your class is determined by the status of your parents at your birth. So if your dad was a plumber when you were born, you are working class. If you then go to university and become a doctor, you are still working class, but your children will be middle class (their status is determined by your occupation at the time of their birth) and they will have been brought up in a middle class household, perhaps privately educated because you can afford it.
There are two ways to move up the class scale - one is to get educated and a good job. The other is to marry up.
A good example of the first is the Home Secretary Savid Javid. He is the son of a Pakistani immigrant who was a bus driver before opening a corner store. Javid then went to university and then became an investment banker, climbing the ranks and earning a seven figure salary. He married his girlfriend from university, Laura King, and then took a pay cut to go into politics, where he quickly climbed again, becoming Home Secretary (one of the four great offices of state).
He has achieved both high income and high status, and his children will be upper middle class in terms of upbringing, wealth and status.
Another example is Kate Middleton. Her father Michael Middleton is a straight forward middle class man. But her mother Carole came from working class roots - she herself was an air hostess, and her ancestors were coal miners and labourers. Carole married "up" when she married Michael Middleton, and the couple started a business and privately educated their children. Thus Kate was raised as middle class though her mother had a working class background.
The daughter also married up - marrying Prince William. All their children will be aristocracy, and their eldest son, George, will be King. So here you have spectacular social mobility, going from working class to monarch within three generations.
The lesson to be learned from both Javid and Kate Middleton is that social mobility is perfectly possible, either through merit or marrying up. Though both require education - Javid wouldn't have got his chance in investment banking without a good degree from Exeter University, and Kate wouldn't have married Prince William if she hadn't been privately educated (giving her the appropriate manners and accent) and hadn't gone to the same university as him (St Andrews).
The moral of the story is that if you want your children to rise in life, educate them.