From the look out of things there is no doubt that $60k is sure before the end of this month
There will always be doubts. I definitely have doubts that bitcoin will reach a price of $60k by the end of the month. After the recent explosive rise in price, bitcoin seems to be trying to catch its breath and doesn't have enough strength for a further continuous run.February reason being that bitcoin has surpasses the price of 50k and headed to $52k if I am not mistakenly and you can correct me if I am wrong.
You're not wrong. Bitcoin reached the $50k level and even reached the ~$52k mark. But the jump in price from ~$40k to $50k doesn't mean that further (in the near future) we will see $60k, $70k, $80k (in series), and so on. You should not expect growth to be linear and constant. No not like this. Cryptomarket always fluctuates and now the price of bitcoin hangs around $50k-52k.Also does this proves the idea of most people saying that bitcoin could hit half a million dollars after the halving by 2025, does this sign proves it right?
Do you mean that at the peaks of this cycle we will see (per) bitcoin for $500k? Well, I don't tink so. Price increases are certainly inevitable, but not to such levels. Leave the expectation of such indicators for subsequent cycles. 
Levels of $100-150k are more realistic.You are correct most place I was wrong and without knowing that bitcoin price doesn't move in series as per say, I couldn't even believe with my eyes that till date we haven't seen bitcoin crosses above 52k level. Even if I was told that by today bitcoin won't cross 53k I will have to doubt with my heart because of it's spontaneous growth initially and its catchy to people eyes which made lot of people invested without even studying the market properly.
Today bitcoin still struggles within 50k to 52k respectively without crossing that level, and from my understanding I can easily say that the selling pressure which is holding bitcoin within this price level, but whenever the buying force supersedes the selling pressure then we can say that bitcoin is bounce to pass this level it's currently.