Soul, Spirit, Atman, Self etc. are words which are used to define subtle nature of ourselves which is invisible & mysterious. But if you say that Give me a proof for the existence of this Soul, how can it be given? Here below is how we can understand this invisible & mysterious Soul.
We in our daily life generally say that, "My Body", "My Mind", "My Senses or Indriyas", "My Pranas", etc. When you address these things as "My" that clearly means that your true Self, the Soul or Atman is entirely different from the Body, the Mind, the Prana and the Senses. The Body & Mind are your servants or instruments. They are as much outside of your true Self, Soul as these outside things as your home, car, bed etc. You are holding this body just as you are holding a long walking stick in your hand. Your true Self, the Soul is the possessor or proprietor of this body. The body is your property or possession. The body, the senses, the mind, etc. are not the Soul, Self or Atman, but belong to it.
Soul is something which is within this body & something which can exist without this body. When soul lives in this physical body then it enjoys the different sensory objects through this body. Soul is indestructible & immortal. It moves from one body to the other. As we are not able to see this Soul due to it's subtle nature so it looks mysterious to us.
Now let's understand about the Soul through the Bhagwat Gita. As per Bhagwat Gita both God (Super soul) and the Individual Souls do exist & It also believes in the incarnation of the individual souls, their delusion and bondage to the cycle of births and deaths because of desire ridden actions. As per this, Soul at death gets released from this body, rises to the higher worlds (heaven etc.) or to the lower wombs (animals, birds etc.) & reincarnates or return to the earth to continue its cycle of births and deaths.
According to Bhagwat Gita, the Soul, or Atman, is indestructible and eternal (Chapter 2.18).
"antavanta ime deha
tasmad yudhyasva bharata" (Bhagwat Gita: Chapter Two verse 18)
"Sri Krishna said: All these bodies pertaining to the indestructible, immeasurable and eternal living entity / soul are spoken of as perishable; therefore, fight, Arjuna."
According to Bhagwat Gita, the Soul, or Atman, neither slays not can it be slain (Chapter 2.19).
"ya enam vetti hantaram
yas chainam manyate hatam
ubhau tau na vijanito
nayam hanti na hanyate" (Bhagwat Gita: Chapter Two verse 19)
"Sri Krishna said: They are both ignorant, he who knows the soul to be capable of killing and he who takes it as killed; for verily the soul neither kills, nor is killed."
According to Bhagwat Gita, the Soul, or Atman, is never born, never dies and after coming into existence never ceases to be. It is nitya (always), shasvatah (permanent) and purana (very ancient) (Chapter 2.20).
"na jayate mriyate va kadacin
nayam bhutva bhavita va na bhuyah
ajo nityah sasvato ’yam purano
na hanyate hanyamane sarire" (Bhagwat Gita: Chapter Two verse 20)
"Sri Krishna said: The soul is never born nor dies at any time. Soul has not come into being, does not come into being, and will not come into being. Soul is unborn, eternal, ever-existing and primeval. Soul is not slain when the body is slain."
According to Bhagwat Gita, the Soul, or Atman, does not suffer & cannot be tainted. At the time of death it does not die, but leaves the body and enters into a new one (Chapter 2.22).
"vasamsi jirnani yatha vihaya
navani grhnati naro ’parani
tatha sarirani vihaya jirnany
anyani samyati navani dehi" (Bhagwat Gita: Chapter Two verse 22)
"Sri Krishna said: As a human being puts on new garments, giving up old ones, the soul similarly accepts new material bodies, giving up the old and useless ones."
According to Bhagwat Gita, the Soul, or Atman has the properties that Weapons cannot pierce it, fire cannot burn it, water cannot moisten it and wind cannot dry it (Chapter 2.23).
"nainam chindanti shastrani
nainam dahati pavakah
na chainam kledayanty apo
na sosayati marutah" (Bhagwat Gita: Chapter Two verse 23)
"Sri Krishna said: The soul can never be cut to pieces by any weapon, nor burned by fire, nor moistened by water, nor withered by the wind."
According to Bhagwat Gita, the Soul, or Atman, is impenetrable, incombustible, all pervading, stable and immobile (Chapter 2.24).
"acchedyo ’yam adahyo ’yam
akledyo ’sosya eva cha
nityah sarva-gatah sthanur
achalo ’yam sanatanah" (Bhagwat Gita: Chapter Two verse 24)
"Sri Krishna said: This individual soul is unbreakable and insoluble, and can be neither burned nor dried. He is everlasting, present everywhere, unchangeable, immovable and eternally the same."
According to Bhagwat Gita, the Soul, or Atman, is invisible, imperceptible and immutable (Chapter 2.25).
"avyakto ’yam acintyo ’yam
avikaryo ’yam ucyate
tasmad evam viditvainam
nanusocitum arhasi" (Bhagwat Gita: Chapter Two verse 25)
"Sri Krishna said: It is said that the soul is invisible, inconceivable and immutable. Knowing this, you should not grieve for the body."
Lord Krishna knew the limitations of human consciousness to gauge the true nature of the inner soul. So in the Bhagwat Gita it concurs that no one can exactly know what a soul is. Someone looks at It with great surprise, another speaks about It with great surprise, some another hears about It with incredulity and yet another after hearing about It knows it not (Chapter 2.29).
"ascharya-vat pasyati kascid enam
ascharya-vad vadati tathaiva chanyah
ascharya-vac chainam anyah srnoti
srutvapy enam veda na chaiva kascit" (Bhagwat Gita: Chapter Two verse 29)
"Sri Krishna said: Some look on the soul as amazing, some describe him as amazing, and some hear of him as amazing, while others, even after hearing about him, cannot understand him at all."
So in this way Soul is a Mysterious thing & can be understood if you have faith in God & in religious scriptures.
Jai Shri Krishna