How do I know that my bitcoins are real I have bought some bitcoins off local bitcoins and stored them in an Electra wallet in my computer. Given that I have now paid hard cash, how am I to determine that these are real?
Enter your Bitcoin address (the one you were paid into) into a public blockchain browser such as https://blockchain.info/. That will show you the relevant entries in the global shared Bitcoin ledger, which confirms that you, as the holder of the private key for that address, are the owner of those funds.

If you lose control of your private key for any reason, then your bitcoins might not be yours for much longer.