How would you deal such situation when everyone in your workplace expects you to fail just because you had mistaken once?

Whether you have done a mistake once or not is the secondary one. If everyone expects you to fail means its a conspiracy and a criminal act. Indeed true these things happen these days especially misusing of social networking site, only innocent are getting suffered because of the group of wrong people creates nonsense or misuses the identity of an innocent and blame him for their wrongful acts and make the innocent suffer the loss or isolate from his contacts, primarily to hide their mistake or wrongful acts. See if10 people wants to show the false thing as true is possible in this world through social networking site through their10 people's network.  This is purely a conspiracy. Many real talented sane people also believes the only these10 people and their network only cause the real innocent is only one person and more even he/she was isolated from the contacts. You can try very vigilant and optimistic only in such a situations, cause these are very difficult situations in anyone's life, but do not lose patience that is the only your arm. Try to deal things with patience(sure lots of patience) cause they do try to provoke you to do mistakes as it is going to advantage for them to show you again wrong. Be careful and vigilant and have patience.  These situation arising not only in work place and  happening every where. If workplace means you can speak to them and make them understand that their approach is wrong and try to resolve things amicably, if not possible better look for a opportunity in elsewhere and settle down may be a better idea, cause it is very difficult to group of wrong people they lack minimum values in them and can go down any low to achieve their ulterior motives or selfish wrongful gains or hide their mistakes.. Thanks.

I think we all have these type people in our lives, who have nothing to do but belittle people and treat them with contempt. Mistakes happen, we all fall that's life. It's about getting up, proving these people wrong. Even if you fail keep trying, your actions will prove them wrong!