Suppose that {x} is a convergent sequence and {y} is such that for any e > 0 there exists M such that |x – y| < e for all n>=M. Does it follow that {y} is convergent?

My approach is as follows:

Since |x – y| < e implies that lim x = lim y.

Since {x} is already convergent and intuitively from above the gap between {x} and {y} must be decreasing as n approaches infinity, can we conclude from here that {y} must also be convergent?