External objects seen in both the dream and waking world are unreal alike. This is what I assume OP needs the proof for.
स्वप्नवृत्तावपि त्वन्तश्चेतसा कल्पितं त्वसत् ।
बहिश्चेतोगृहीतं सद्दृष्टं वैतथ्यमेतयोः ॥ ९ ॥
जाग्रद्वृत्तावपि त्वन्तश्चेतसा कल्पितंत्वसत् ।
बहिश्चेतो गृहीतं सद्युक्तं वैतथ्यमेतयोः ॥ १० ॥
-Mandukya Karika 2.9-2.10
9-10. In dream, also, what is imagined within by the mind is illusory
and what is cognized outside (by the mind) appears to be real. But (in
truth) both these are known to be unreal. Similarly, in the waking
state, also, what is imagined within by the mind is illusory; and what
is experienced outside (by the mind) appears to be real. But in fact,
both should be rationally held to be unreal.
To prove the fact that external objects in waking world are also unreal lets look at who cognizes these external objects.
कल्पयत्यात्मनाऽऽत्मानमात्मा देवः स्वमायया |
स एव बुध्यते भेदानिति वेदान्तनिश्चयः ॥ १२ ॥
-Mandukya Karika 2.12
- Ātman, the self-luminous, through the power of his own Māyā, imagines in himself by himself (all the objects that the subject experiences within or without). He alone is the cognizer of the objects (so created). This is the decision of the Vedānta.
I will prove now that external objects do not exist at any point of time, and only Atman exists all the time. This will prove the above hypothesis. First let me define Atman and external objects.
Atman - Something that has no attribute and exist all the time
External objects - Ordinary objects which have birth-death and exist
for some time
Firstly notice that immortal can never turn mortal and vice-versa. This proves that Atman can never give rise to external objects.
न भवत्यमृतं मर्त्यं न मर्त्यममृतं तथा ।
प्रकृतेरन्यथाभावो न कथंचिद्भविष्यति ॥ २१ ॥
-Mandukya Karika 3.21
- The immortal cannot become mortal, nor can the mortal ever become immortal. For, it is never possible for a thing to change its nature.
Lets talk about reality of external objects now. They have birth and death, i.e. they come into being for some time and then go out of existence after some time.
Is the thing which causes their birth real or unreal? An unreal thing can never be the cause of real external object.
But production, by the non-existent, of any thing either in reality or
in illusion is not possible. For we know nothing like it in our
experience. As the son of a barren woman is not seen to be born either
really or through Māyā, the theory of the non-existence of things is
in truth1 untenable.
-Shankara Bhashya on MK 3.28
So, what causes the external object? It cannot be Atman, as we decided earlier. Neither can it be created out of its own or something non existent. The last option is that it came out from another external object.
But we cannot certify the reality of external object by assuming the reality of another external object as its cause because that itself is what needs to be proved. This is example of infinite regress
Hence proved that reality cannot be associated with external objects and only unborn nondual Atman is real.
फलादुत्पद्यमानस्सन्न ते हेतुः प्रसिध्यति ।
अप्रसिद्धः कथं हेतुः फलमुत्पादयिष्यति ॥ १७ ॥
- Your cause cannot be established if it be produced from the effect. How can the causey which is itself not established, give birth
to the effect?
Shankara Bhashya (commentary)
How can there be no causal relation? It
is thus replied:—The cause1 cannot have a definite existence if it is
to be born of an effect which is, itself, yet unborn, and therefore
which is non-existent like the horns of a hare. How2 can the cause
contemplated by you, which is, itself, indefinite and which is
non-existent like the horns of a hare, produce an effect? Two things
which are mutually dependent upon each other for their production and
which are like3 the horns of a hare, cannot be related as cause and
effect or in4 any other way.
Objection: If external objects do not exist, why do we feel pain, color, etc. when an external object is present and not without it?
(Shankara Bhashya on MK 2.24) Cognition of sound, etc., is not possible without objects. For, such
experience is always produced by a cause. In1 the absence of such
(external) object, the variety and multiplicity of experiences such as
sound, touch, colour, viz., blue, yellow, red, etc., would not have
existed. But the varieties are not non-existent, for these are
directly perceived by all. Hence, because: the variety of manifold
experiences exist, it is necessary to admit the existence—as supported
by the system of the opposite school—of external objects which are
outside the ideas of the perceiving subject. The subjective knowledge
has one characteristic alone, i.e., it is of the very nature of
illumination. It does not admit of any variety within itself. The
variety of experiences of colour, such as blueness, yellowness, etc.,
cannot possibly be explained, by merely imagining a variety in the
subjective knowledge, without admitting variety of external objects
which are the substratum of these multiple colours. In other words, no
variety of colour is possible in a (white) crystal without its
coming-in contact with such adjuncts as the external objects which
possess such colours as blueness, etc. For this additional reason also
one is forced to admit the existence of external object,—supported by
the Scripture of the opposite school,—an object which is external to
the knowledge (of the perceiving subject): Misery caused by burns,
etc., is experienced by all. Such pain as is caused by burns, etc.,
would not have been felt in the absence of the fire, etc., which is
the cause of the burns and which exists independent of the knowledge
(of the perceiving subject). But such pain is experienced by all.
Hence,3 we think that external objects do exist. It is not reasonable
to conclude that such pain is caused by mere subjective knowledge.
For,4 such misery is not found elsewhere.
(Shankara Bhashya 4.25) To this objection, we reply as follows:—We admit that you posit a cause of the subjective experience
on account of such arguments as the existence of the variety (in the
objective world) and because of the experience of pain. Stick for a
while to your argument that reason demands that an external object
should exist to produce a subjective impression.
let us know what you (Advaitin) are going to say next.
the jar, etc., posited by you as the cause, that is to say, the cause
of the subjective impression, are not, according to us, the external
cause, the substratum (of the impression); nor are they the cause for
our experiences of variety.
(Reply)—We say so from
the standpoint of the true nature of Reality. When the true nature of
clay is known a jar does not exist apart from the clay as exists a
buffalo in entire independence of a horse. Nor does cloth exist apart
from the thread in it. Similarly the threads have no existence apart
from the fibres. If we thus proceed to find out the true nature of the
thing, by going from one cause to another, till language or the object
denoted by the language fails us, we do not still find any (final)
MK stands for Mandukya Karika. You can read the book for free here.