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Hi Flar,
Nice to see a new face and asking the right question.
As silu_nz says, the BR cushion is to handle variance that comes with poker, you want to be able to survive a downswing where you play good poker but the luck kills.
And trust me when I say you don't know what this really means. There are top players who have gone 100,000 hands + without winning (or even losing); I've barely played 100k hands in total...
If I understand you right you've got around $100 which is 20 buy ins of $5. You will minimise your risk by buying in for $5 at the $0.02/$0.05 tables.
If you're finding it's going too slow then why not multi-table? Add a second, or third table or fourth until you can play winning poker like that.
If you're reasonably confident that you're better than a level then winning 5ptbb/100 is not unrealistic which means it'll take you 20,000 hands to double up 20 buy-ins; and if that sounds like a lot of hands then you're not yet on board with what grinding is.
There's nothing wrong with taking the odd risk, but you have to be reasonably smart about it.
I mean you could, say, play $5 buy ins for a bit and if you make $20 (so now you're $120) you make one of your tabels a $10 ($0.10bb) table, but if you lose the $20, you move back down and try again, if you make it to $140 you add a second and so on until you're 4-tabling the higher limit.
You many find it all goes great, in which case good luck to you. But most people don't have it that easy and prepare yourself for it now: it's going to take a loooot of hands to get where you want to go (if you want to get up past $100NL) so you may as well get in the habit of grinding now.
When you hear people say that there's no difference between $2NL and $100NL they're either wrong (happens), looking at it from a mid-stakes+ perspective (in which case the large differences seem relatively small) or they're saying something more subtle:
A good game plan that wins at $2NL will win at $100NL. But take a look at the number of low-stakes players trying to break into/past $100NL and you'll see a lot of struggle. Because your leaks get exposed more and more as you move up stakes. A winning $2NL style may not win at $10NL, let alone $100NL.
It sounds like you've already committed yourself to grinding your way up. So good luck, keep focused and try to remove impatience and focus on improving your skills so that when your BR grows you're ready to use it.
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