//If I am conscious while I am alive, how does that show that death won't end my consciousness?//

Ok, fast forward to Dec 31st of this year and let's say you woke up that morning with a case of total amnesia--you remember nothing about your life, not a memory or anything, you're like a newborn infant. When did it seem that life began for you? Just that morning. Now, could this happen to you? Since experiencing always happens only in the present moment and you are conscious now and have memories, then no, it cannot happen to you. If you were going to get complete amnesia on Dec 31st, you could have no memories until Dec 31st or later. Since you are conscious now and do have memories, you will not get total amnesia on the Dec 31st or at any time in the future or you would not be conscious now.

Now, replace total ammesia with death (which is a form of total amnesia if it wipes out your consciousness). Remember: your experiencing is always in the "now," the present moment, and not in the past or future. Are you experiencing now? Yes. But if death extinguished your consciousness, the "now" cannot occur until after the amnesia occurs. But since death supposedly wipes out consciousness then "now" can't occur at the point either. It never occurs. You are never conscious. But since you recall all your experiences, you ARE conscious--contradiction. Death does not extinguish consciousness.

If you are conscious for even one second or a microsecond, then you are conscious for eternity.

Another way to look at it is to suppose that event X occurs now and you observe it. For you to be conscious, you must recall X at some future point which we'll call X'. X' is what binds future and present together as well as present and past. Without X', X never happened as far as you are concerned because you don't remember it. So if consciousness is wiped out before X' occurs, then you cannot be conscious now.