Hello there. I'm A Hacker since 6 Years.
Let Me First Explain Something :
how Did he Get Your Password ?
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If you're Using the Same Router , then It's Easy To Spy on your Computer .
there are Many Ways, the First One Is Called " Sniffing the Packets" , Which Starts by Installing a special Program Called "Wireshark"
The Second Method is A little bit Harder, It Consists of Several Things :
* the Main Program ((Prorat v2.0 / Bifrost / Brmoda English Version"))
* the Server (it's a virus > Trojan Horse / Backdoor/ Worms)
*The Victim's IP address ( 192.168.0.100 For Example)
Don't Leave your Computer without a Password.
Don't Leave your Administrator Password Empty
Go to Start > run > cmd > net user Administrator *
They will ask you for a new password ... Type something.
download a Program Called "Hostpost Shield Free Version with ads"
http://www.hotspotshield.com
It's a VPN > Virtual Private Network.
Install it , and Secure your Connection.
now your ISP > Internet Service provider won't be able to spy on you.
You Stepfather Won't be able to sniff your packets (to Spy on you and Get the passwords)
Download anti MalwareBytes :
http://www.malwarebytes.org
and scan your Computer for any Viruses.
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Ok now about your Question , How to Prove that Your StepFather Accessed your E-mail Illegally without your Permission, and that he Sent and deleted Some Messages ...
UNFORTUNATELY there's no Direct way.
There's one Solution :
Download Wireshark , and Sniff his Packets , Get the password of his e-mail address, and Log-in , Take some ScreenShots, and Save them. That's How You will have a clear Image that Shows The Received E-mails (Which are The Details)
If you're Using the Same Router, Then You Must Consider the Advanced-Spying-Method :
Send him a Server , Connect to his Computer Via that Virus, once Done , Monitor everything, and Wait Until you Get The Passwords.
P.S : You can Also Retrieve His Files From his Hard-Disk-Drive.
Here are The Steps:
*download Prorat ( any version 1.9 or 2.0 are the best )
here's a tutorial :
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http://hackinginception.blogspot.com/2011/11/hack-remote-computers-using-prorat.html
and here's a tutorial Video
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http://www.youtube.com/watch?v=MVhAd7h-zPw
If you want to Sniff the packets (Method 1 )
*here's a list of videos :
http://www.youtube.com/results?search_query=sniff+packets+with+Wireshark
and a tutorial :
http://www.brighthub.com/computing/smb-security/articles/33432.aspx
http://vishnuvalentino.com/hacking-tutorial/how-to-sniff-http-post-password-via-network-using-wireshark-network-analyzer/
>>>>>>>>>>>. and As I told you , You can Prove everything If you get a Direct access to his Computer/Machine and to his E-mail account.
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Edited
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If you have a G-mail account , then you can look for this sentence, its Found in the bottom of the page
"Last account activity: 45 minutes ago on this computer. Details" > Example
The details link will open a pop up that will have the IP address and timestamp for your last few logins
There's also an option to add a secondary login credential for gmail if you're really paranoid: http://googleenterprise.blogspot.com/2010/09/more-secure-cloud-for-millions-of.html
if you have a Yahoo E-mail
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After logging into your yahoo account, go to your "Account Information" page from the top right (below your name);
You will be again asked for your password. When the page opens, click on the link "View your recent sign-in activity" under Sign-In and Security section;
A new page will be opened, on which you can view the details of signing 'time', 'date', 'location' and 'type of login';
And after you find any kind of misbehavior with your account, you can contact the Customer Care
if you want more instructions. (below on the same page)
===> THUMBS DOWN ?
Jealousy Kills People .. it Really Kills !!!