Prove that a space is simply connected So I've been working through this book (A Basic Course in Algebraic Topology, by William Massey) in preparation for an Algebraic Topology course I'm going to take soon, and I ran into trouble with this exercise.

Let ${U_i}$ be an open covering of the space X having the following properties:
(a) There exists a point $x_0$ such that $x_0$ ϵ $U_i$ for all i.
(b) Each $U_i$ is simply connected.
(c) If i≠j, then $U_i$ ∩ $U_j$ is arcwise connected.
Prove that X is simply connected.

The book hints at proving that any loop  f: I$\to$X (where I = [0,1] ) based at $x_0$ is trivial. I know I should use the fact that {$f^{-1}$($U_i$)} is an open covering of the compact space I, which means that there exists a finite subcovering of I. However, I am not sure how to proceed from there. Any hint at all would be appreciated. Thanks in advance.
 A: Take a path $\phi\colon I \to X$ starting and ending at $x_0$. From the Lebesgue number lemma (Proof of the Lebesgue number lemma) for the covering $\phi^{-1}(U_i)$ of $I$  we conclude that there exists $n \ne 1$ so that $[\frac{k-1}{n}, \frac{k}{n}]$ is inside some $\phi^{-1}(U_i)$, that is, $\phi([\frac{k-1}{n}, \frac{k}{n}] )\subset U_{i_k}$. For $1< k <n$ , $k$ natural, the point $\phi(\frac{k}{n})$ is in  $U_{i_k}\cap U_{i_{k+1}}$, and so is $x_0$. Take a path $\psi_k \colon I \to U_{i_k}\cap U_{i_{k+1}}$   connecting $x_0$ and $\phi(\frac{k}{n})$. 
Here are the steps : 
Decompose $\phi$ into a composition of $n$ paths $\phi_k$ that are basically the restrictions of $\phi$ to $[\frac{k-1}{n}, \frac{k}{n}]$
Show that 
$$\phi_1 \star \ldots \star \phi_n  \simeq \phi_1 \star \psi_1^{-1} \star \psi_1  \star \phi_2 \star \psi_2^{-1}  \ldots \star \phi_n $$ 
For each $k$ the path $\psi_{k-1}\star \phi_k \star \psi_k$ is a closed look in $U_{i_k}$ and therefore homotopic to a trivial path. 
Piece together to get a homotopy from the loop $\phi$ to the constant loop at $x_0$. 
There are some alternate, equivalent ways to define the path composition and the equivalence of paths, some making certain proofs easier -- compare with the path composition in the book by Crowell and Fox, Knot Theory http://www.maths.ed.ac.uk/~aar/papers/crowfox.pdf. 
