The problem asks to show that if the space Y is path connected, then [I,Y], the set of homotopy classes of maps I into Y where I = [0,1], consists of a single element. What I tried to do is take two arbitrary continuous maps f,g: I -> Y and show that they are homotopic. For each s ε I, f(s) and g(s) are elements of Y. Thus, by our assumption, there exists some path, call it ps, such that ps(0)=f(s) and ps(1)=g(s). Define F: I X I -> Y by the equation F(s,t)=ps(t)for each s ε I. Then, F(s,0)=ps(0) = f(s) and F(s,1)=ps(1)=g(s)for each s. I have since learned that this is incorrect as the correct approach would be to take an arbitrary continous map and show that it is homotopic to a constant map. My question is regarding where my argument failed. Any help/guidance would be great! Thank you!