HELP! my iPhone won't start without wall charger
so i had this iphone 2g running 3.1.3 on snowbreeze
this is almost 3 years with me,this past months,the battery life of it shortens,from having a day to 18 hrs to 6 hrs
so last week i charged my phone 50% charged and working on.,then after some minutes i checked it and it was off
so i removed the cable and start it normally by pressing the on button at the top,but it does not boot,
i thought its a software problem,but after leaving it charging it in an hour,i turned it on and it started normally but after the battery drained in an hour
then i charged it hoping it will start,but even the battery charging icon does'nt show,it's a black screen,so i think my software problem thought is real,
but after searching on the web, i saw a tip explaining this:
"press the on button together with the home button for 8 sec. then let go of the buttons and start it normally"
so i tried it and my phone shows the apple logo and after some minutes it became black,so i charged it again and do the same steps then when the apple logo flashes,i quickly removed the charger then the screen goes out SLOWLY so i think i have a battery problem but im not sure with it,so please suggest what should do,should the replacement battery will solve this?