1st semester teaching embarassment / disaster

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Recently got hired to teach part time and was assigned Real Analysis to which I was more than excited.

However in the first week when I tried to explain how the set [0,1] contains and infinite amount of real numbers I was immediately met with skepticism by a student.

He proceeded to press me on the matter and since I was new I became visibly anxious, slurring my words which was further intensified when I deduced that the set was not just infinite, but uncountably infinite to which I was met with skepticism as another student joined in pressing me.

At this point I was visibly uncomfortable as it is a very bad sign to be doubted not just once but in a consecutive matter to the entire class.

The final straw was when I, using the same reasoning arrived at (0,1) having the same cardinality as the entirely of R to which a different student proceeded to walk out of the classroom.

By the end, I was sweating bullets and I am now dreading coming back to lecture on monday. How do new adjuncts deal with this type of anxiety ? is it normal to be pressed this much at first ? if my students are now skeptical is there anything I can do to win their trust back?

If the anxiety Is preventing me from effectively lecturing will I have to resign?

I am absolutely terrified as I have to touch on cauchy sequences and equivalence classes very soon and I already know what their reaction will be like.

I think I will have to talk to the chair of the department for possible options.