user4820: Part 1: 
I was studying some of the Buddha's teaching, when I encountered this notion that Buddhism does not recognize 'Atman' or soul. Buddha did not believe in notion of 'Self'; He did not believe that there was any permanent/in-destructible component in Humans. If there is no permanent/common part between the birth cycles, How can Re-incarnation be justified/proven?
edit: 
Part 2 : 
If Buddhism does not believe in "Self", How can there be Nirvana for one?

user471: Part 1:
The answer comes from Milindapanha 3.5.5. This means that rebirth without transmigration of the soul from one place to another, takes place like a transfer of "information".

The king asked: "Venerable Nagasena, is it so that one does not
  transmigrate and one is reborn?" "Yes, your majesty, one does not
  transmigrate and one is reborn." "How, venerable Nagasena, is it that
  one does not transmigrate and one is reborn? Give me an analogy."
  "Just as, your majesty, if someone kindled one lamp from another, is
  it indeed so, your majesty, that the lamp would transmigrate from the
  other lamp?" "Certainly not, venerable sir." "Indeed just so, your
  majesty, one does not transmigrate and one is reborn." "Give me
  another analogy." "Do you remember, your majesty, when you were a boy
  learning some verse from a teacher?" "Yes, venerable sir." "Your
  majesty, did this verse transmigrate from the teacher?" "Certainly
  not, venerable sir." "Indeed just so, your majesty, one does not
  transmigrate and one is reborn."

Here's another quote by Ven. Narada Mahathera about the transfer of "information" using the analogy of radio waves picked up by a radio set:

Rebirth takes place immediately, irrespective of the place of birth,
  just as an electromagnetic wave, projected into space, is immediately
  reproduced in a receiving radio set. Rebirth of the mental flux is
  also instantaneous and leaves no room whatever for any intermediate
  state (antarabhava). Pure Buddhism does not support the belief that a
  spirit of the deceased person takes lodgement in some temporary state
  until it finds a suitable place for its "reincarnation."

I summarized in another answer (based on the Theravada tradition) that:

Every person's habitual thoughts and actions (karma) shapes his
  "inclination of awareness", that becomes the last state of mind at
  death, and this is the "information" that gets transferred without
  carrier, the way flame is transferred from candle to candle, or radio
  waves are transferred from one radio to another.

Part 2:
The question is not clear to me. With Nirvana, one ceases to appear in a new life again in the future.

user4808: There is a difference between re-incarnation and re-birth.  The Dhamma-Vinaya teaches rebirth based upon paticcasamuppada [dependent arising]. The usual religious epistemology has humans having a body, mind (conscious), and spirit (soul).  The Gotama Buddha was pragmatic in his logis  and the Dhamma-Vinaya is empirical.  Since the existence of a soul or an inherent Self can not be evinced, we are left with the pragmatic solution of dependent arising as taught by the Enlightened One.  Since the body becomes decrepit and dies what we are left with is consciousness that becomes attached to the next re-birth.  It is a consciousness devoid of the last personality but still encumbered with some of the past ignorance/dispositions (and occasionally some memory of past lives).
Nibbana is not a place; like the mythical heaven of religions.  Nibbana is the release, the freedom from samsara; the cycle of re-births.  Everyone and anyone can attain nibbana.  Once the dispositions that are the result of greed, hatred and etc. are dissipated.  This is actually a much deeper topic to discuss than can be done here.    

4820: <p>Part 1: <br>
I was studying some of the Buddha's teaching, when I encountered this notion that Buddhism does not recognize 'Atman' or soul. Buddha did not believe in notion of 'Self'; He did not believe that there was any permanent/in-destructible component in Humans. If there is no permanent/common part between the birth cycles, How can Re-incarnation be justified/proven?</p>

<p>edit: <br>
Part 2 : <br>
If Buddhism does not believe in "Self", How can there be Nirvana for one?</p>
 471: <h2>Part 1:</h2>

<p>The answer comes from <a href="http://www.accesstoinsight.org/tipitaka/kn/miln/miln.3x.kell.html#miln-3-5-05" rel="nofollow noreferrer">Milindapanha 3.5.5</a>. This means that rebirth without transmigration of the soul from one place to another, takes place like a transfer of "information".</p>

<blockquote>
  <p>The king asked: "Venerable Nagasena, is it so that one does not
  transmigrate and one is reborn?" "Yes, your majesty, one does not
  transmigrate and one is reborn." "How, venerable Nagasena, is it that
  one does not transmigrate and one is reborn? Give me an analogy."
  "Just as, your majesty, if someone kindled one lamp from another, is
  it indeed so, your majesty, that the lamp would transmigrate from the
  other lamp?" "Certainly not, venerable sir." "Indeed just so, your
  majesty, one does not transmigrate and one is reborn." "Give me
  another analogy." "Do you remember, your majesty, when you were a boy
  learning some verse from a teacher?" "Yes, venerable sir." "Your
  majesty, did this verse transmigrate from the teacher?" "Certainly
  not, venerable sir." "Indeed just so, your majesty, one does not
  transmigrate and one is reborn."</p>
</blockquote>

<p>Here's another <a href="http://www.buddhism.org/Sutras/BuddhaTeachings/page_28.html" rel="nofollow noreferrer">quote</a> by Ven. Narada Mahathera about the transfer of "information" using the analogy of radio waves picked up by a radio set:</p>

<blockquote>
  <p>Rebirth takes place immediately, irrespective of the place of birth,
  just as an electromagnetic wave, projected into space, is immediately
  reproduced in a receiving radio set. Rebirth of the mental flux is
  also instantaneous and leaves no room whatever for any intermediate
  state (antarabhava). Pure Buddhism does not support the belief that a
  spirit of the deceased person takes lodgement in some temporary state
  until it finds a suitable place for its "reincarnation."</p>
</blockquote>

<p>I summarized in another <a href="https://buddhism.stackexchange.com/a/7792/471">answer</a> (based on the Theravada tradition) that:</p>

<blockquote>
  <p>Every person's habitual thoughts and actions (karma) shapes his
  "inclination of awareness", that becomes the last state of mind at
  death, and this is the "information" that gets transferred without
  carrier, the way flame is transferred from candle to candle, or radio
  waves are transferred from one radio to another.</p>
</blockquote>

<h2>Part 2:</h2>

<p>The question is not clear to me. With Nirvana, one ceases to appear in a new life again in the future.</p>
 4808: <p>There is a difference between re-incarnation and re-birth.  The Dhamma-Vinaya teaches rebirth based upon paticcasamuppada [dependent arising]. The usual religious epistemology has humans having a body, mind (conscious), and spirit (soul).  The Gotama Buddha was pragmatic in his logis  and the Dhamma-Vinaya is empirical.  Since the existence of a soul or an inherent Self can not be evinced, we are left with the pragmatic solution of dependent arising as taught by the Enlightened One.  Since the body becomes decrepit and dies what we are left with is consciousness that becomes attached to the next re-birth.  It is a consciousness devoid of the last personality but still encumbered with some of the past ignorance/dispositions (and occasionally some memory of past lives).</p>

<p>Nibbana is not a place; like the mythical heaven of religions.  Nibbana is the release, the freedom from samsara; the cycle of re-births.  Everyone and anyone can attain nibbana.  Once the dispositions that are the result of greed, hatred and etc. are dissipated.  This is actually a much deeper topic to discuss than can be done here.    </p>
