Q:

Is there any proof that diphthongs exist?

I was always taught that a word contains as many syllables as it has vowels. By definition,  a vowel is a sound that produces a syllable.
On the other hand, in English phonology, by definition, diphthong is two adjacent vowels which produce one syllable. I did not hear about diphthongs until I read some literature about English phonology.  
I am aware about long vowels (which produce one syllable) and vowel hiatus (two consecutive vowels) which produce two syllables. So what is the proof that English diphthongs are actually two vowels which produce one syllable, as opposed to long vowels, hiatus, or a vowel followed by a consonant? 

A:

In thinking about this it's important to distinguish between phonetics and phonology.
Phonetically a diphthong is a sequence of two vowel targets, wherein the tongue starts at one vowel position and moves to another. For this reason it may sometimes be described as a combination of vowel and glide, but it's best to understand the articulatory facts of what it involves, movement from one tongue 'vowel' position to another, as opposed to the steady-state articulation found in simple 'monopthongal' vowels.
The evidence for this can come from simply hearing the vowel movement, as well as from various kinds of imaging such as spectrograms, ultrasounds, x-rays, etc.
Phonologically such an articulation can function as a single phoneme, as in the diphthongs of English. But a sequence of two phonemes consisting of vowel + vowel, or vowel + glide, can (eg in fast speech) be phonetically identical to a diphthong.
As for English diphthongs, a standard phonemic analysis of English will reveal a set of vowel phonemes. In every variety of English that I'm aware of some of those vowel phonemes will be realised as diphthongs, involving an articulatory movement from one vowel target to another, as revealed by the kinds of evidence already mentioned. Some of the evidence of the phonemic status of these diphthongs in English includes:

standard minimal sets: 'beet, bit, bet, bait, bite, boat, bout,
bought, boot' etc
Diphthong as sole syllabic nucleus, eg 'how', 'boat', 'I'
Participation in same tense-lax contrasts as monopthongal vowels, eg
'divine-divinity', cf. 'serene-serenity'

A:

Defining a diphthong as a sequence of two vowels may not be correct.    A diphthong is a sequence comprising a vowel and a glide, at least, according to the Summer Institute of Linguistics:   http://www-01.sil.org/linguistics/GlossaryOflinguisticTerms/WhatIsADiphthong.htm 
A glide, or semivowel, is defined at https://en.wikipedia.org/wiki/Glide_%28linguistics%29 as “a sound, such as English /w/ or /j/, that is phonetically similar to a vowel sound but functions as the syllable boundary rather than as the nucleus of a syllable.”   In other words, a sound that is like a vowel, but non-syllabic.   
When the glide comes first in the diphthong, we have words such as British “new” [nju], in which [j] as in "yes" [jɛs] is the glide.    
When the glide comes last in the diphthong, we have words like English “aisle” [aɪ̯l], in which [ɪ̯] is the glide.   
Note the little curve symbol under the <ɪ>, yielding <ɪ̯>.    That little curve, called an inverted breve, means that the vowel sound is non-syllabic, i.e. that it’s a glide.
Incidentally, vowels aren’t the only sounds that can be syllable nuclei.   Consider syllabic consonants, such as the [̯̯̯̯l̩] in “bottle” [bɑdl̩].     Also see the examples mentioned with this definition of syllabic consonants:   http://www-01.sil.org/linguistics/GlossaryOfLinguisticTerms/WhatIsASyllabicConsonant.htm 

