First, Scotland can’t stay in the EU after Brexit because Scotland isn’t a member. The UK is a member. Scotland isn’t. Greenland was part of Denmark and therefore in the EU. When it separated from Denmark, it ceased to be an EU member.
Sturgeon is selling the lie that she believes Scotland can stay in a union it never belonged to by becoming an independent country. It’s not going to happen because it would require the agreement of all 27 countries to admit Scotland if it applied after independence. Spain has already said it would veto this, fearing an independent Catalonia.
Secondly an independent Scotland joining the EU would be required to adopt the Euro. Sturgeon wants to keep sterling, but the EU wouldn’t allow it. It’s a requirement of joining the EU that you take the Euro.
Thirdly the economy of Scotland isn’t strong enough to stand on its own. It gets money from England and it gets money from the EU. After Brexit, the UK would have more money to spend internally as it currently gives money to the EU part of which they give back to Scotland. The UK could match that quite easily after Brexit. Put simply, Scotland would be worse off outside the UK even if it was admitted to the EU.
In any case, Scotland on its own wouldn’t meet the economic criteria for membership – although I accept that those criteria have been fudged before, in Greece for example.
Finally, being part of the EU isn’t independence. If you accept your laws and economy being dictated by Brussels, then you’re not independent. You just swapped one union for another.
Sturgeon knows all of this, but doesn’t want the voters in Scotland to know it. For her it’s independence at any price – including not really being independent at all…